Cross-sectional Dependence in Idiosyncratic Volatility

abstractThis paper introduces an econometric framework for analyzing cross-sectional dependence in the idiosyncratic volatilities of assets using high frequency data. We first consider the estimation of standard measures of dependence in the idiosyncratic volatilities such as covariances and correlations. Naive estimators of these measures are biased due to the use of the error-laden estimates of idiosyncratic volatilities. We provide bias-corrected estimators and the relevant asymptotic theory. Next, we introduce an idiosyncratic volatility factor model, in which we decompose the variation in idiosyncratic volatilities into two parts: the variation related to the systematic factors such as the market volatility, and the residual variation. Again, naive estimators of the decomposition are biased, and we provide bias-corrected estimators. We also provide the asymptotic theory that allows us to test whether the residual (non-systematic) components of the idiosyncratic volatilities exhibit cross-sectional dependence. We apply our methodology to the S&P 100 index constituents, and document strong cross-sectional dependence in their idiosyncratic volatilities. We consider two different sets of idiosyncratic volatility factors, and find that neither can fully account for the cross-sectional dependence in idiosyncratic volatilities. For each model, we map out the network of dependencies in residual (non-systematic) idiosyncratic volatilities across all stocks. Keywords: factor model, systematic risk, networks of risk, residual idiosyncratic volatility, (co-)volatility of volatility, high frequency data. \\ \newline JEL Codes: C58, C22, C14, G11.

assumptionSuppose $Y$ is an It\^{o} semimartingale on a filtered space $(\Omega ,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$, \begin{equation} Y_{t}=Y_{0}+\int_{0}^{t}b_{s}ds+\int_{0}^{t}{\Greekmath 011B} _{s}dW_{s}+\int_{0}^{t}\int_{E}{\Greekmath 010E} (s,z){\Greekmath 0116} (ds,dz), \end{equation} where $W$ is a $d^{W}$-dimensional Brownian motion ($d^{W}\geq d$) and ${\Greekmath 0116} $ is a Poisson random measure on $\mathbb{R}_{+}\times E$, with $E$ an auxiliary Polish space with intensity measure ${\Greekmath 0117} (dt,dz)=dt\otimes {\Greekmath 0115} (dz)$ for some ${\Greekmath 011B} $-finite measure ${\Greekmath 0115} $ on $E$. The process $ b_{t} $ is $\mathbb{R}^{d}$-valued optional, ${\Greekmath 011B} _{t}$ is $\mathbb{R} ^{d}\times \mathbb{R}^{d^{W}}$-valued, and ${\Greekmath 010E} ={\Greekmath 010E} (w,t,z)$ is a predictable $\mathbb{R}^{d}$ -valued function on $\Omega \times \mathbb{R} _{+}\times E$. Moreover, $\Vert {\Greekmath 010E} (w,t\wedge {{\Greekmath 011C} _{m}(w)},z)\Vert \wedge 1\leq \Gamma _{m}(z)$, for all (w,t,z), where (${\Greekmath 011C} _{m}$) is a localizing sequence of stopping times and, for some $r\in \lbrack 0,1/2)$, the function $\Gamma _{m}$ on $E$ satisfies $\int_{E}\Gamma _{m}(z)^{r}{\Greekmath 0115} (dz)<\infty $. The spot volatility matrix of $Y$ is then defined as $C_{t}={\Greekmath 011B} _{t}{\Greekmath 011B} _{t}^{\top }$. We assume that $C_{t}$ is an It\^{o} semimartingale,\footnote{ Note that $\widetilde{{\Greekmath 011B} }_{s}=(\widetilde{{\Greekmath 011B} }_{s}^{gh,m})$ is $ (d\times d\times d^{W})$-dimensional and $\widetilde{{\Greekmath 011B} }_{s}dW_{s}$ is $ (d\times d)$-dimensional with $(\widetilde{{\Greekmath 011B} }_{s}dW_{s})^{gh}= \sum_{m=1}^{d^{W}}\widetilde{{\Greekmath 011B} }_{s}^{gh,m}dW_{s}^{m}$.} \begin{equation} C_{t}=C_{0}+\int_{0}^{t}\widetilde{b}_{s}ds+\int_{0}^{t}\widetilde{{\Greekmath 011B} } _{s}dW_{s}+J_{t}^{{\Greekmath 011B} }, \end{equation} where $\widetilde{b}$ is $\mathbb{R}^{d}\times \mathbb{R}^{d}$-valued optional, and $J_{t}^{{\Greekmath 011B} }$ is a finite activity jump process. $C_{t}$ takes values in the space $\mathcal{M}_{d}$ consisting of $d\times d$ positive definite matrices. For a sequence of convex compact subsets $( \mathcal{K}_{m})_{m\geq 1}$ of $\mathcal{M}_{d}$, $C_{t}\in \mathcal{K}_{m}$ for all $t\leq {\Greekmath 011C} _{m}$.

assumption$\widetilde{{\Greekmath 011B} }_{t}$ is a continuous It\^{o} semimartingale with its characteristics satisfying the same requirements as that of $C_{t}-J_{t}^{{\Greekmath 011B} }$.

theoremLet $\widehat{[H_{r}(C),G_{r}(C)]_{T}^{c}}$ denote either $ \widehat{[H_{r}(C),G_{r}(C)]_{T}^{c}}^{AN}$ or $\widehat{ [H_{r}(C),G_{r}(C)]_{T}^{c}}^{LIN}$ defined in equations ((ref)) and ((ref)), where $H_{r}$ and $G_{r}$ are three times differentiable real-valued functions, for $r=1,\ldots ,{\Greekmath 0114} $. Suppose Assumptions (ref) and (ref) hold. Fix $k_{n}={\Greekmath 0112} \Delta _{n}^{-1/2}$ for some ${\Greekmath 0112} \in (0,\infty )$. Set $u_{n}\asymp \Delta _{n}^{{\Greekmath 0124} }$ with $\frac{2{\Greekmath 0124}^{\prime }+9}{4\left( 5-r\right) } < {\Greekmath 0124} <\frac{1}{2}$, and $ u_{n}^{\prime }\asymp \Delta _{n}^{{\Greekmath 0124} ^{\prime }}$ with $0 < {\Greekmath 0124} ^{\prime }<\min \left( \frac{1}{2}-r,\frac{1}{8}\right) $ . Then, as $ \Delta _{n}\rightarrow 0$, \begin{equation} \Delta _{n}^{-1/4}\left( \begin{array}{c} \widehat{\lbrack H_{1}(C),G_{1}(C)]_{T}^{c}}-[H_{1}(C),G_{1}(C)]_{T}^{c} \\ \ldots \\ \widehat{\lbrack H_{{\Greekmath 0114} }(C),G_{{\Greekmath 0114} }(C)]_{T}^{c}}-[H_{{\Greekmath 0114} }(C),G_{{\Greekmath 0114} }(C)]_{T}^{c} \end{array} \right) \overset{\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{L-s}}{\longrightarrow }MN(0,\Sigma _{T}). \end{equation} Let $\Sigma _{T}^{r,s}$ be the $\left( \Sigma _{T}\right) _{r,s}$ element of the ${\Greekmath 0114} \times {\Greekmath 0114} $ matrix $\Sigma _{T}$. We have \begin{align*} & \Sigma _{T}^{r,s}=\Sigma _{T}^{r,s,(1)}+\Sigma _{T}^{r,s,(2)}+\Sigma _{T}^{r,s,(3)}, \\ & \Sigma _{T}^{r,s,(1)}=\frac{6}{{\Greekmath 0112} ^{3}}\sum_{g,h,a,b=1}^{d} \sum_{j,k,l,m=1}^{d}\int_{0}^{T}\big(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s}(C_{s})\big)\Big[ C_{t}(gh,jk)C_{t}(ab,lm) \\ & +C_{t}(ab,jk)C_{t}(gh,lm)\Big]dt, \\ & \Sigma _{T}^{r,s,(2)}=\frac{151{\Greekmath 0112} }{140}\sum_{g,h,a,b=1}^{d} \sum_{j,k,l,m=1}^{d}\int_{0}^{T}\big(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s}(C_{t})\big)\Big[\overline{C} _{t}^{gh,jk}\overline{C}_{t}^{ab,lm} \\ & +\overline{C}_{t}^{ab,jk}\overline{C}_{t}^{gh,lm}\Big]dt, \\ & \Sigma _{T}^{r,s,(3)}=\frac{3}{2{\Greekmath 0112} }\sum_{g,h,a,b=1}^{d} \sum_{j,k,l,m=1}^{d}\int_{0}^{T}\big(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s}(C_{t})\big)\Big[C_{t}(gh,jk) \overline{C}_{t}^{ab,lm} \\ & +C_{t}(ab,lm)\overline{C}_{t}^{gh,jk}+C_{t}(gh,lm)\overline{C} _{t}^{ab,jk}+C_{t}(ab,jk)\overline{C}_{t}^{gh,lm}\Big]dt, \end{align*} with \begin{equation*} C_{t}(gh,jk)=C_{gj,t}C_{hk,t}+C_{gk,t}C_{hj,t}. \end{equation*}

theoremSuppose the assumptions of Theorem (ref) hold. Then, as $\Delta _{n}\rightarrow 0$, \begin{align} & \frac{6}{{\Greekmath 0112} ^{3}}\widehat{\Omega }_{T}^{r,s,(1)}\overset{\mathbb{P}}{ \longrightarrow }\Sigma _{T}^{r,s,(1)}, \\ & \frac{3}{2{\Greekmath 0112} }[\widehat{\Omega }_{T}^{r,s,(3)}-\frac{6}{{\Greekmath 0112} } \widehat{\Omega }_{T}^{r,s,(1)}]\overset{\mathbb{P}}{\longrightarrow }\Sigma _{T}^{r,s,(3)},\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and} \\ & \frac{151{\Greekmath 0112} }{140}\frac{9}{4{\Greekmath 0112} ^{2}}[\widehat{\Omega } _{T}^{r,s,(2)}+\frac{4}{{\Greekmath 0112} ^{2}}\widehat{\Omega }_{T}^{r,s,(1)}-\frac{4}{ 3}\widehat{\Omega }_{T}^{r,s,(3)}]\overset{\mathbb{P}}{\longrightarrow } \Sigma _{T}^{r,s,(2)}. \end{align}

corollaryLet $\widehat{[H_{r}(C),G_{r}(C)]_{T}^{c}}$ denote either $\widehat{ [H_{r}(C),G_{r}(C)]_{T}^{c}}^{AN}$ or $\widehat{[H_{r}(C),G_{r}(C)]_{T}^{c}} ^{LIN}$ defined in equations ((ref)) and ((ref)). Suppose the assumptions of Theorem (ref) hold. Then, as $\Delta _{n}\rightarrow 0$, \begin{equation} \Delta _{n}^{-1/4}\ \widehat{\Sigma }_{T}^{-1/2}\left( \begin{array}{l} \widehat{\lbrack H_{1}(C),G_{1}(C)]_{T}^{c}}-[H_{1}(C),G_{1}(C)]_{T}^{c} \\ \vdots \\ \widehat{\lbrack H_{{\Greekmath 0114} }(C),G_{{\Greekmath 0114} }(C)]_{T}^{c}}-[H_{{\Greekmath 0114} }(C),G_{{\Greekmath 0114} }(C)]_{T}^{c} \end{array} \right) \overset{L}{\longrightarrow }N(0,I_{{\Greekmath 0114} }). \end{equation}

sidewaystable[thb] \begin{center} \begin{tabular}{lcccccccc} \hline\hline & & \multicolumn{3}{c}{\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{CAPM}} & & \multicolumn{3}{c}{\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{FF3 Model}}\\ $|\widehat{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Corr}}(Z_i,Z_j)|$ & & Pairs & Avg\,$|\widehat{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Corr}}(Z_i,Z_j)|$ & Avg\,$\widehat{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Corr}}\left(C_{Zi},C_{Zj} \right)$ & & Pairs & Avg\,$|\widehat{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Corr}}(Z_i,Z_j)|$ & Avg\,$\widehat{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Corr}}\left(C_{Zi},C_{Zj} \right)$ \\ \cline{1-1} \cline{3-5} \cline{7-9} $<0.6$ & & 5356 & 0.045 & 0.347 & & 5356 & 0.045 & 0.347\\ $<0.5$ & & 5354 & 0.045 & 0.347 & & 5353 & 0.045 & 0.347\\ $<0.4$ & & 5334 & 0.044 & 0.346 & & 5335 & 0.044 & 0.346\\ $<0.3$ & & 5300 & 0.042 & 0.344 & & 5300 & 0.042 & 0.345\\ $<0.2$ & & 5236 & 0.039 & 0.343 & & 5236 & 0.039 & 0.344\\ $<0.1$ & & 4925 & 0.033 & 0.338 & & 4928 & 0.033 & 0.339\\ $<0.075$ & & 4642 & 0.030 & 0.333 & & 4647 & 0.030 & 0.333\\ $<0.050$ & & 3873 & 0.024 & 0.320 & & 3895 & 0.024 & 0.320\\ $<0.025$ & & 2049 & 0.013 & 0.296 & & 2044 & 0.013 & 0.296\\ $<0.010$ & & 757 & 0.005 & 0.293 & & 748 & 0.005 & 0.293\\ $<0.005$ & & 374 & 0.003 & 0.297 & & 373 & 0.002 & 0.296\\ \hline\hline \end{tabular} \caption{Each row in this table describes the subset of pairs of stocks with $|\widehat{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Corr}(Z_i,Z_j)}|$ below a threshold in column one. The table considers two R-FMs: the left panel defines the IdioVol with respect to CAPM, and the right panel defines the IdioVol with respect to the three-factor Fama-French model. In both cases, the market volatility is the only IdioVol factor. Each panel reports three quantities for the given subset of pairs: the number of pairs, average absolute pairwise correlation in idiosyncratic returns, and average pairwise correlation between IdioVols. } \end{center}

itemize• the IdioVol factor loading of the first stock, ${\Greekmath 010D} _{Z1}$, • the contribution of the market volatility to the variation of the IdioVol of the first stock $R_{Z1}^{2,\textit{IdioVol-FM}}$, • the correlation between the Idiosyncratic Volatilities of stocks 1 and 2, $Corr\left( C_{Z1},C_{Z2}\right) $, • the correlation between the residual Idiosyncratic Volatilities, $ Corr\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $.

sidewaystable\begin{tabular*}{1.0\textwidth}{l|@{\extracolsep{\fill}}cccccccccccc} \hline\hline \rule{0pt}{15pt}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{} & \multicolumn{4}{c}{LIN} & \multicolumn{4}{c}{AN} & \multicolumn{4}{c}{Naive} \\ $\widehat{{\Greekmath 0112} }$ & 1.5 & 2 & 2.5 & 3 & 1.5 & 2 & 2.5 & 3 & 1.5 & 2 & 2.5 & 3 \\ \hline & & & & & & & & & & & & \\ & \multicolumn{12}{c}{Median Bias} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & -0.007 & -0.004 & -0.005 & -0.011 & -0.032 & -0.027 & -0.025 & -0.028 & -0.257 & -0.230 & -0.209 & -0.177 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & -0.153 & -0.138 & -0.127 & -0.115 & -0.146 & -0.132 & -0.121 & -0.110 & -0.484 & -0.465 & -0.448 & -0.417 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & -0.129 & -0.104 & -0.086 & -0.059 & -0.147 & -0.118 & -0.100 & -0.070 & -0.342 & -0.334 & -0.325 & -0.307 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & -0.089 & -0.064 & -0.045 & -0.018 & -0.109 & -0.082 & -0.061 & -0.029 & -0.245 & -0.239 & -0.232 & -0.218 \\ & \multicolumn{12}{c}{\rule{0pt}{15pt}IQR} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & 0.173 & 0.157 & 0.141 & 0.118 & 0.173 & 0.154 & 0.140 & 0.118 & 0.079 & 0.078 & 0.078 & 0.076 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & 0.180 & 0.166 & 0.154 & 0.133 & 0.201 & 0.185 & 0.170 & 0.141 & 0.040 & 0.042 & 0.044 & 0.046 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & 0.279 & 0.257 & 0.238 & 0.211 & 0.321 & 0.289 & 0.266 & 0.229 & 0.039 & 0.041 & 0.043 & 0.048 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & 0.330 & 0.304 & 0.280 & 0.249 & 0.381 & 0.344 & 0.311 & 0.273 & 0.040 & 0.042 & 0.044 & 0.049 \\ & \multicolumn{12}{c}{\rule{0pt}{15pt}RMSE} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & 0.130 & 0.116 & 0.105 & 0.090 & 0.132 & 0.118 & 0.108 & 0.093 & 0.263 & 0.238 & 0.217 & 0.185 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & 0.206 & 0.185 & 0.170 & 0.150 & 0.242 & 0.192 & 0.174 & 0.152 & 0.484 & 0.466 & 0.449 & 0.418 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & 0.257 & 0.226 & 0.203 & 0.169 & 0.309 & 0.260 & 0.229 & 0.187 & 0.343 & 0.335 & 0.327 & 0.309 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & 0.300 & 0.261 & 0.235 & 0.199 & 0.394 & 0.309 & 0.266 & 0.213 & 0.247 & 0.241 & 0.234 & 0.221 \\ \\ \hline\hline \end{tabular*} {\caption{Finite sample properties of our estimators using 10 years of data sampled at 5 minutes. The true values are ${\Greekmath 010D}_{Z1}=0.450$, $R^{2,\textit{IdioVol-FM}}_{Z1}=0.342$, $Corr \left( C_{Z1},C_{Z2} \right)=0.523$, $Corr\left(C^{resid}_{Z1},C^{resid}_{Z2} \right)=0.424$. Model 2. }}

sidewaystable\begin{tabular*}{1.0\textwidth}{l|@{\extracolsep{\fill}}cccccccccccc} \hline\hline \rule{0pt}{15pt}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{} & \multicolumn{4}{c}{LIN} & \multicolumn{4}{c}{AN} & \multicolumn{4}{c}{Naive} \\ $\widehat{{\Greekmath 0112} }$ & 1.5 & 2 & 2.5 & 3 & 1.5 & 2 & 2.5 & 3 & 1.5 & 2 & 2.5 & 3 \\ \hline & & & & & & & & & & & & \\ & \multicolumn{12}{c}{Median Bias} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & -0.034 & -0.029 & -0.022 & -0.013 & -0.052 & -0.044 & -0.036 & -0.025 & -0.304 & -0.295 & -0.275 & -0.267 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & -0.140 & -0.123 & -0.109 & -0.086 & -0.135 & -0.117 & -0.103 & -0.080 & -0.496 & -0.492 & -0.477 & -0.473 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & -0.146 & -0.128 & -0.114 & -0.091 & -0.163 & -0.143 & -0.129 & -0.104 & -0.327 & -0.323 & -0.321 & -0.317 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & -0.118 & -0.105 & -0.095 & -0.076 & -0.138 & -0.123 & -0.111 & -0.092 & -0.220 & -0.216 & -0.216 & -0.212 \\ & \multicolumn{12}{c}{\rule{0pt}{15pt}IQR} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & 0.147 & 0.132 & 0.118 & 0.100 & 0.146 & 0.131 & 0.117 & 0.099 & 0.062 & 0.062 & 0.063 & 0.063 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & 0.165 & 0.148 & 0.137 & 0.119 & 0.176 & 0.158 & 0.145 & 0.125 & 0.032 & 0.032 & 0.034 & 0.034 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & 0.260 & 0.232 & 0.209 & 0.175 & 0.287 & 0.249 & 0.224 & 0.188 & 0.032 & 0.032 & 0.033 & 0.033 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & 0.312 & 0.280 & 0.254 & 0.211 & 0.341 & 0.303 & 0.273 & 0.225 & 0.032 & 0.032 & 0.033 & 0.033 \\ & \multicolumn{12}{c}{\rule{0pt}{15pt}RMSE} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & 0.115 & 0.102 & 0.091 & 0.076 & 0.121 & 0.106 & 0.095 & 0.078 & 0.307 & 0.299 & 0.279 & 0.271 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & 0.192 & 0.165 & 0.147 & 0.121 & 0.198 & 0.168 & 0.148 & 0.121 & 0.496 & 0.493 & 0.478 & 0.474 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & 0.251 & 0.220 & 0.196 & 0.162 & 0.283 & 0.243 & 0.215 & 0.177 & 0.328 & 0.324 & 0.322 & 0.318 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & 0.291 & 0.249 & 0.221 & 0.182 & 0.760 & 0.279 & 0.245 & 0.199 & 0.221 & 0.218 & 0.218 & 0.214 \\ \\ \hline\hline \end{tabular*} {\caption{Finite sample properties of our estimators using 10 years of data sampled at 1 minute. The true values are ${\Greekmath 010D}_{Z1}=0.450$, $R^{2,\textit{IdioVol-FM}}_{Z1}=0.336$, $Corr \left( C_{Z1},C_{Z2} \right)=0.514$, $Corr\left(C^{resid}_{Z1},C^{resid}_{Z2} \right)=0.408$. Model 2.}}

sidewaystable\begin{tabular*}{1.0\textwidth}{l|@{\extracolsep{\fill}}cccccccccccc} \hline\hline \rule{0pt}{15pt}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{} & \multicolumn{4}{c}{LIN} & \multicolumn{4}{c}{AN} & \multicolumn{4}{c}{Naive} \\ $\widehat{{\Greekmath 0112} }$ & 1.5 & 2 & 2.5 & 3 & 1.5 & 2 & 2.5 & 3 & 1.5 & 2 & 2.5 & 3 \\ \hline & & & & & & & & & & & & \\ & \multicolumn{12}{c}{Median Bias} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & -0.075 & -0.072 & -0.068 & -0.061 & -0.096 & -0.089 & -0.083 & -0.075 & -0.323 & -0.315 & -0.299 & -0.291 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & -0.183 & -0.169 & -0.155 & -0.139 & -0.183 & -0.169 & -0.156 & -0.137 & -0.500 & -0.496 & -0.484 & -0.480 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & -0.187 & -0.169 & -0.161 & -0.145 & -0.214 & -0.194 & -0.185 & -0.166 & -0.321 & -0.316 & -0.317 & -0.313 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & -0.144 & -0.128 & -0.125 & -0.116 & -0.167 & -0.155 & -0.146 & -0.139 & -0.209 & -0.205 & -0.207 & -0.202 \\ & \multicolumn{12}{c}{\rule{0pt}{15pt}IQR} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & 0.229 & 0.205 & 0.184 & 0.154 & 0.225 & 0.202 & 0.184 & 0.154 & 0.092 & 0.092 & 0.093 & 0.093 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & 0.246 & 0.223 & 0.206 & 0.177 & 0.265 & 0.238 & 0.218 & 0.187 & 0.047 & 0.047 & 0.049 & 0.049 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & 0.407 & 0.357 & 0.325 & 0.281 & 0.453 & 0.394 & 0.354 & 0.299 & 0.047 & 0.046 & 0.049 & 0.048 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & 0.475 & 0.419 & 0.387 & 0.324 & 0.529 & 0.462 & 0.420 & 0.352 & 0.047 & 0.047 & 0.049 & 0.049 \\ & \multicolumn{12}{c}{\rule{0pt}{15pt}RMSE} \\ \rule{0pt}{15pt}$\widehat{{\Greekmath 010D} }_{Z1}$ & 0.184 & 0.165 & 0.150 & 0.127 & 0.192 & 0.172 & 0.156 & 0.134 & 0.330 & 0.321 & 0.307 & 0.298 \\ \rule{0pt}{15pt}$\widehat{R}_{Z1}^{2,\textit{IdioVol-FM}}$ & 0.330 & 0.240 & 0.218 & 0.188 & 0.420 & 0.246 & 0.225 & 0.192 & 0.501 & 0.497 & 0.486 & 0.482 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1},C_{Z2}\right) $ & 0.409 & 0.342 & 0.307 & 0.260 & 0.500 & 0.388 & 0.345 & 0.285 & 0.322 & 0.318 & 0.319 & 0.314 \\ \rule{0pt}{15pt}$\widehat{Corr}\left( C_{Z1}^{resid},C_{Z2}^{resid}\right) $ & 0.510 & 0.399 & 0.355 & 0.287 & 0.813 & 0.481 & 0.417 & 0.323 & 0.212 & 0.207 & 0.209 & 0.205 \\ \\ \hline\hline \end{tabular*} {\caption{Finite sample properties of our estimators using 5 years of data sampled at 1 minute. The true values are ${\Greekmath 010D}_{Z1}=0.450$, $R^{2,\textit{IdioVol-FM}}_{Z1}=0.35$, $Corr \left( C_{Z1},C_{Z2} \right)=0.517$, $Corr \left( C^{resid}_{Z1},C^{resid}_{Z2} \right)=0.417$. Model 2.}}

appendix{ Appendix} \setcounter{theorem}{0} \setcounter{section}{0} \setcounter{subsection}{0} \setcounter{equation}{0} \numberwithin{figure}{section} \numberwithin{table}{section} \setcounter{page}{1} Sections (ref)-(ref) contain all proofs. Section (ref) contains some numerical implementation details. Section (ref) contains additional figures for the empirical application. The proofs are organised as follows. Section (ref) introduces additional notation. Section (ref) presents auxiliary theorems and lemmas used to prove Theorems (ref) and (ref) in the main paper. Section (ref) proves Theorem (ref). Section (ref) proves Theorem (ref). Section (ref) collects the proofs of the auxiliary results of Section (ref). \section{Notation for Proofs} Our notation is similar to that of the proofs of jacodrosenbaum-sqrtn whenever possible. Throughout, we denote by $K$ a generic constant, which may change from line to line. We let by convention $\sum_{i=a}^{a^{\prime }}=0$ when $a> a^{\prime }$. For simplicity, we omit the subscript $r$ for results involving only one object with this subscript. By the usual localization argument, there exists a ${\Greekmath 0119} $ -integrable function $J$ on $E$ and a constant such that the stochastic processes in equations ((ref)) and ((ref)) satisfy \begin{equation} \Vert b\Vert ,\Vert \widetilde{b}\Vert ,\Vert c\Vert ,\Vert \widetilde{c} \Vert ,J\leq A,\Vert {\Greekmath 010E} (w,t,z)\Vert ^{r}\leq J(z). \end{equation} We set \begin{equation*} \mathcal{F}_{i}^{n}=\mathcal{F}_{i \Delta _{n}}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, } C_{i}^{n}=C_{i\Delta _{n}}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, } \overline{C}_{i}^{n}=\overline{C}_{i\Delta _{n}} \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and } \widehat{C}_{i}^{n}=\widehat{C}_{i\Delta _{n}}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.} \end{equation*} For any c\`{a}dl\`{a}g bounded process $Z$, we set \begin{align*} & {\Greekmath 0111} _{t,s}(Z)=\sqrt{\mathbb{E}\Big(\sup_{0<u\leq s}\Vert Z_{t+u}-Z_{t}\Vert ^{2}|\mathcal{F}_{t}\Big)}, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \\ & {\Greekmath 0111} _{i,j}^{n}(Z)=\sqrt{\mathbb{E}\Big(\sup_{0\leq u\leq j\Delta _{n}}\Vert Z_{(i-1)\Delta _{n}+u}-Z_{(i-1)\Delta _{n}}\Vert ^{2}|\mathcal{F} _{(i-1)\Delta _{n}}\Big)}. \end{align*} For convenience, we decompose $Y_{t}$ as \begin{equation*} Y_{t}=Y_{0}+Y_{t}^{\prime }+\sum_{s\leq t}\Delta Y_{s}. \end{equation*} where $Y_{t}^{\prime }=\int_{0}^{t}b_{s}^{^{\prime }}ds+\int_{0}^{t}{\Greekmath 011B} _{s}dW_{s}$ and $b_{t}^{\prime }=b_{t}-\int {\Greekmath 010E} (t,z)1_{\{\Vert {\Greekmath 010E} (t,z)\Vert \leq 1\}}{\Greekmath 0119} (dz)$.\newline Let $\widehat{C}_{i}^{\prime n}$ be the local estimator of the spot variance of the unobservable process $Y^{\prime }$, i.e., \begin{equation} \widehat{C}_{i}^{\prime n}=\frac{1}{k_{n}\Delta _{n}}\sum_{u=0}^{k_{n}-1}( \Delta _{i+u}^{n}Y^{\prime })(\Delta _{i+u}^{n}Y)^{\prime \top }=(\widehat{C} _{i}^{\prime n,gh})_{1\leq g,h\leq d}. \end{equation} There is no price jump truncation applied in the definition of $\widehat{C} _{i}^{\prime n}$ since the process $Y^{\prime }$ is continuous. Hence, it is more convenient to work with $\widehat{C}_{i}^{\prime n}$ rather than $ \widehat{C}_{i}^{n}$ ($=\widehat{C}_{i\Delta _{n}}$, defined in equation ((ref))). We also define \begin{equation} {\Greekmath 010B} _{i}^{n} =(\Delta _{i}^{n}Y^{\prime })(\Delta _{i}^{n}Y^{\prime})^{\top } - C_{(i-1)\Delta _{n}}\Delta _{n}, {\Greekmath 0117} _{i}^{n} =\widehat{C}_{i}^{^{\prime }n}-C_{(i-1)\Delta _n}, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} {\Greekmath 0115}_{i}^{n} =\widehat{C}_{i+k_{n}}^{^{\prime }n}-\widehat{C}_{i}^{^{\prime }n}, \end{equation} which satisfy \begin{equation} {\Greekmath 0117} _{i}^{n} = \frac{1}{k_{n}\Delta _{n}}\sum_{j=0}^{k_{n}-1}({\Greekmath 010B}_{i+j}^{n} +(C_{(i+j-1)\Delta _n}-C_{(i-1)\Delta _n})\Delta _{n}) \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} {\Greekmath 0115} _{i}^{n} = {\Greekmath 0117} _{i+k_{n}}-{\Greekmath 0117} _{i}^{n} +C_{(i+k_n-1)\Delta _n}-C_{(i-1)\Delta _n}. \end{equation} The following multidimensional quantities will be used in the sequel \begin{center} \begin{tabular}{ll} ${\Greekmath 0110} (1)_{i}^{n}=\frac{1}{\Delta _{n}}\Delta _{i}^{n}Y^{\prime }(\Delta _{i}^{n}Y^{\prime })^{\top }-C_{i-1}^{n},$ & ${\Greekmath 0110} (2)_{i}^{n}=\Delta _{i}^{n}c,$ \\ ${\Greekmath 0110} ^{\prime }(u)_{i}^{n}=\mathbb{E}({\Greekmath 0110} (u)_{i}^{n}|\mathcal{F} _{i-1}^{n}),$ & ${\Greekmath 0110} ^{\prime \prime }(u)_{i}^{n}={\Greekmath 0110} (u)_{i}^{n}-{\Greekmath 0110} ^{\prime }(u)_{i}^{n},$ \\ ${\Greekmath 0110} ^{r}(u)_{i}^{n}=\Big({\Greekmath 0110} ^{r}(u)_{i}^{n,gh}\Big)_{1\leq g,h\leq d}$ & $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{with }r=^{\prime }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ or }^{\prime \prime }.$ \end{tabular} \end{center} For $1\leq g,h\leq d$ and $u,v=1,2$, define \begin{equation*} {\Greekmath 011A} _{gh}(u,v)_{i}^{n}=\sum_{m=1}^{2k_{n}-1}{\Greekmath 0115} (u,v)_{m}^{n}{\Greekmath 0110} _{gh}(u)_{i-m}^{n}. \end{equation*} We also define, for $m\in \{0,\ldots ,2k_{n}-1\}$ and $j,l\in \mathbb{Z}$, \begin{equation*} {\Greekmath 0122} (1)_{m}^{n}= \begin{cases} -1 & if0\leq m<k_{n} \\ +1 & ifk_{n}\leq m<2k_{n}, \end{cases} ,\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\Greekmath 0122} (2)_{m}^{n}=\sum_{q=m+1}^{2k_{n}-1}{\Greekmath 0122} (1)_{q}^{n}=(m+1)\wedge (2k_{n}-m-1), \end{equation*} For any $u,v,m,u^{\prime },v^{\prime }$, we set \begin{equation*} z_{u,v}^{n}= \begin{cases} 1/\Delta _{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=v=1 \\ 1 & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}, \end{cases} \end{equation*} \begin{align*} {\Greekmath 0115} (u,v;m)_{j,l}^{n}& =\frac{3}{2k_{n}^{3}}\sum_{q=0\vee (j-m)}^{(l-m-1)\vee (2k_{n}-m-1)}{\Greekmath 0122} (u)_{q}^{n}{\Greekmath 0122} (u)_{q+m}^{n},{\Greekmath 0115} (u,v)_{m}^{n}={\Greekmath 0115} (u,v;m)_{0,2k_{n}}^{n}, \\ M(u,v;u^{\prime },v^{\prime })_{n}& =z_{u,v}^{n}z_{u^{\prime },v^{\prime }}^{n}\sum_{m=1}^{2k_{n}-1}{\Greekmath 0115} (u,v)_{m}^{n}{\Greekmath 0115} (u^{\prime },v^{\prime })_{m}^{n}. \end{align*} We also need some notation for volatility jumps. Denote by $N_{s}$ the number of jumps in $C$ from time $0$ to $s$. Let \begin{eqnarray} L\left( n\right) &=&\left\{ i=k_{n}+1,k_{n}+2,...:N_{\left( i+3\right) k_{n}\Delta _{n}}-N_{\left( i-1\right) k_{n}\Delta _{n}}=0\right\} , \notag \\ L\left( n,T\right) &=&\left\{ i=1,2,...,\left[ T/\Delta _{n}\right] -3k_{n}+1\right\} \cap L\left( n\right) , \\ L^{\prime }\left( n,T\right) &=&\left\{ i=1,2,...,\left[ T/\Delta _{n}\right] :i-2k_{n}\in L\left( n,T\right) \right\} , \notag \\ \overline{L}\left( n,T\right) &=&\left\{ i=1,2,...,\left[ T/\Delta _{n} \right] -3k_{n}+1\right\} \backslash L\left( n\right) . \notag \end{eqnarray} Additionally, set \begin{align} \overline{A11}(H,gh,u;G,ab,v)_{T}^{n}& = \frac{3}{2k_{n}^{3}}\sum_{i\in L^{\prime }\left( n,T\right) }\Big(\sum_{j=0}^{2k_{n}-1}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}\Big)(\partial _{gh}H\partial _{ab}G)(C_{(i-2k_n-1)\Delta_n}){\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab} \notag \\ & = {\Greekmath 0115} (u,v)_{0}^{n}\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{(i-2k_n-1)\Delta_n}){\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}, \end{align} and \begin{align} \overline{A12}(H,gh,u;G,ab,v)_{T}^{n}& =\frac{3}{2k_{n}^{3}}\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{(i-2k_n-1)\Delta_n})\sum_{m=1}^{(i-1)\wedge (2k_{n}-1)}\sum_{j=0}^{(2k_{n}-m-1)}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n} \notag \\ & \times {\Greekmath 0110} _{gh}(u)_{i-m}^{n}{\Greekmath 0110} _{ab}(v)_{i}^{n}. \end{align} Denote by ${\Greekmath 0123} _{i}^{AN}$ and ${\Greekmath 0123} _{i}^{LIN}$ the $i^{th}$ summand of $\widehat{\left[ H(C),G(C)\right] _{T}^{c}}^{AN}$ and $\widehat{ \left[ H(C),G(C)\right] _{T}^{c}}^{LIN}$, without the volatility jump truncation, so they satisfy \begin{eqnarray} \widehat{\left[ H(C),G(C)\right] _{T}^{c}}^{AN} &=&\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{AN}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} },\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and} \\ \widehat{\left[ H(C),G(C)\right] _{T}^{c}}^{LIN} &=&\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{LIN}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} } \end{eqnarray} Let ${\Greekmath 0123} _{i}$ be either ${\Greekmath 0123} _{i}^{LIN}$ or ${\Greekmath 0123} _{i}^{AN} $. \section{Auxiliary Lemmas and Theorems} This section presents useful auxiliary results, which are used in the proofs of Theorems (ref) and (ref). The results of this section are proved in Section (ref) below. First, we explain why we can assume, without loss of generality, that the derivatives of functions $H_{r}$ and $G_{r}$ are bounded, for $r=1,\ldots ,{\Greekmath 0114} $. Assumptions of Theorem (ref) imply Lemma 2 of litodorovtauchen17-adaptive. Therefore, we can assume that the variables $ \widehat{C}_{i\Delta _{n}}$ are bounded, uniformly over $i\in \left\{ 0,..., \left[ T/\Delta _{n}\right] -k_{n}+1\right\} $, with probability approaching one. Using the spatial localization argument of litodorovtauchen-dependencies, which in turn uses the spatial localization argument of litodorovtauchen17-adaptive, we can assume that $H_{r}$ and $G_{r}$ are compactly supported without loss of generality. Hence, the derivatives of functions $H_{r}$ and $G_{r}$ are bounded, for $r=1,\ldots ,{\Greekmath 0114} $. We start with two auxiliary theorems for volatility jump truncation. \begin{theorem} Under the assumptions of Theorem (ref), we have \begin{equation} \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}=o_{p}\left( \Delta _{n}^{1/4}\right) . \notag \end{equation} \end{theorem} \begin{theorem} Under the assumptions of Theorem (ref), we have \begin{equation} \sum_{i\in \overline{L}\left( n,T\right) }{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }=o_{p}\left( \Delta _{n}^{1/4}\right) . \notag \end{equation} \end{theorem} Theorems (ref) and (ref) allow us to focus on the simpler leading term $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}$ instead of the original estimator(s) $\sum_{i=k_{n}+1}^{[T/ \Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }$ for the remaining proofs. Our next theorem shows negligibility of price jump truncation. \begin{theorem} Let ${\Greekmath 0123} _{i}^{\prime LIN}$ and ${\Greekmath 0123} _{i}^{\prime AN}$ be the modifications of ${\Greekmath 0123} _{i}^{LIN}$ and $ {\Greekmath 0123} _{i}^{AN}$ obtained by replacing $\widehat{C}_{i}^{n}$ by $ \widehat{C}_{i}^{^{\prime }n}$ in the definition of ${\Greekmath 0123} _{i}^{LIN}$ and ${\Greekmath 0123} _{i}^{AN}$ in equations ((ref)) and ((ref)). Under the assumptions of Theorem (ref), we have \begin{align} \Delta _{n}^{-1/4}\Big(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{LIN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime LIN}\Big)& \overset{\mathbb{P}}{\longrightarrow }0 \notag \\ \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\Delta _{n}^{-1/4}\Big(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{AN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime AN}\Big)& \overset{\mathbb{P}}{\longrightarrow }0. \end{align} \end{theorem} Theorem (ref) allows, in particular, to focus on the derivation of the asymptotic distributions of $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime LIN}$ and $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime AN}$. The next theorem connects the LIN and AN versions of these quantities. To state the theorem, define \begin{align} {\Greekmath 0123} _{i}^{\left( A\right) }& =\frac{3}{2k_{n}}\sum_{g,h,a,b=1}^{d} \Bigg(\Big(\partial _{gh}H\partial _{ab}G\big) (C_{(i-1)\Delta_n})\Big[(\widehat{C} _{i+k_{n}}^{^{\prime }n,gh}-\widehat{C}_{i}^{^{\prime }n,gh})(\widehat{C} _{i+k_{n}}^{^{\prime }n,ab}-\widehat{C}_{i}^{^{\prime }n,ab}) \\ & -\frac{2}{k_{n}}(\widehat{C}_{i}^{^{\prime }n,ga}\widehat{C}_{i}^{^{\prime }n,hb}+\widehat{C}_{i}^{^{\prime }n,gb}\widehat{C}_{i}^{^{\prime }n,ha})\Big] \Bigg). \notag \end{align} where superscript $\left( A\right) $ stands for \textquotedblleft approximated". For simplicity, we do not index the above quantity by a prime although it depends on $\widehat{C}_{i}^{^{\prime }n}$ instead of $\widehat{C}_{i}^{n}$. \begin{theorem} Under the assumptions of Theorem (ref), we have \begin{align} & \Delta _{n}^{-1/4}\Big(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime LIN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }\Big)\overset{\mathbb{P}}{\longrightarrow }0\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \notag \\ & \Delta _{n}^{-1/4}\Big(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime AN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }\Big)\overset{\mathbb{P}}{\longrightarrow }0, \end{align} where ${\Greekmath 0123} _{i}^{\left( A\right) }$ is defined in equation ((ref)). \end{theorem} Theorem (ref) shows that the leading terms of the the two estimators of $\widehat{[H(C),G(C)]}_{T}^{c}$, $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime LIN}$ and $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime AN}$ can be approximated by a certain quantity with an error of approximation of order smaller than $\Delta _{n}^{-1/4}$. Now, we decompose the approximated estimator as follows \begin{equation} {\Greekmath 0123} _{i}^{\left( A\right) }={\Greekmath 0123} _{i}^{\left( A1\right) }-{\Greekmath 0123} _{i}^{\left( A2\right) }, \end{equation} with \begin{equation*} {\Greekmath 0123} _{i}^{\left( A1\right) }=\frac{3}{2k_{n}}\sum_{g,h,a,b=1}^{d}\big( \partial _{gh}H\partial _{ab}G\big)(C_{i-1}^{n})(\widehat{C} _{i+k_{n}}^{^{\prime }n,gh}-\widehat{C}_{i}^{^{\prime }n,gh})(\widehat{C} _{i+k_{n}}^{^{\prime }n,ab}-\widehat{C}_{i}^{^{\prime }n,ab}), \end{equation*} and \begin{equation*} {\Greekmath 0123} _{i}^{\left( A2\right) }=\frac{3}{k_{n}^{2}}\sum_{g,h,a,b=1}^{d} \big(\partial _{gh}H\partial _{ab}G\big)(C_{i-1}^{n})(\widehat{C} _{i}^{^{\prime }n,ga}\widehat{C}_{i}^{^{\prime }n,hb}+\widehat{C} _{i}^{^{\prime }n,gb}\widehat{C}_{i}^{^{\prime }n,ha}). \end{equation*} The following theorem holds: \begin{theorem} Under the assumptions of Theorem (ref), we have \begin{align*} & \frac{1}{\Delta _{n}^{1/4}}\Bigg(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A1\right) }-\sum_{g,h,a,b=1}^{d}\sum_{u,v=1}^{2}\overline{A11} (H,gh,u;G,ab,v)_{T}^{n}+\overline{A12}(H,gh,u;G,ab,v)_{T}^{n} \\ & +\overline{A12}(G,ab,v;H,gh,u)_{T}^{n}\Bigg)\overset{\mathbb{P }}{\Longrightarrow }0. \end{align*} \end{theorem} \begin{lemma} For any c\`{a}dl\`{a}g bounded process $Z$, for all $ t,s>0 $, $j,k\geq 0$, set ${\Greekmath 0111}_{t,s}={\Greekmath 0111}_{t,s}(Z)$. Then, \begin{align*} &\Delta_n \mathbb{E}\Bigg(\sum_{i=1}^{[t/\Delta_n]} {\Greekmath 0111}_{i,k_n}\Bigg) \longrightarrow 0,\Delta_n \mathbb{E}\Bigg(\sum_{i=1}^{[t/ \Delta_n]} {\Greekmath 0111}_{i,2k_n}\Bigg)\longrightarrow 0, \\ & \mathbb{E}\Bigg({\Greekmath 0111}_{i+j,k}|\mathcal{F}_i^n\Bigg)\leq {\Greekmath 0111}_{i,j+k}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\Delta_n \mathbb{E}\Bigg(\sum_{i=1}^{[t/\Delta_n]} {\Greekmath 0111}_{i,4k_n}\Bigg)\longrightarrow 0. \end{align*} \end{lemma} \begin{lemma} Let $Z$ be a continuous It\^o process with drift $b_t^Z$ and spot variance process $C_t^{Z}$, and set ${\Greekmath 0111}_{t,s}={\Greekmath 0111}_{t,s}(b^Z,c^Z)$. Then, the following bounds hold: \begin{align} &\Big|\mathbb{E}(Z_t\Big|\mathcal{F}_0)-tb_0^Z\Big|\leq Kt{\Greekmath 0111}_{0,t} \notag \\ &\Big|\mathbb{E}(Z_t^{j}Z_t^{k}-tC_0^{Z,jk}\Big|\mathcal{F}_0)\Big|\leq Kt^{3/2}(\sqrt{\Delta_n}+{\Greekmath 0111}_{0,t}) \notag \\ &\Big|\mathbb{E}\big((Z_t^{j}Z_t^{k}-tC_0^{Z,jk})(C_t^{Z,lm}-C_0^{Z,lm})\Big| \mathcal{F}_0\big)\Big|\leq Kt^2 \notag \\ &\Big|\mathbb{E}(Z_t^{j}Z_t^{k}Z_t^{l}Z_t^{m}\Big|\mathcal{F} _0)- \Delta_n^2(C_0^{Z,jk}C_0^{Z,lm}+C_0^{Z,jl}C_0^{Z,km}+C_0^{Z,jm}C_0^{Z,kl}) \Big|\leq Kt^{5/2} \notag \\ &\Big|\mathbb{E}(Z_t^{j}Z_t^{k}Z_t^{l}\Big|\mathcal{F}_0)\Big|\leq Kt^2 \notag \\ &\Big|\mathbb{E}(\prod_{l=1}^6 Z_t^{j_l}\Big|\mathcal{F}_0)-\frac{\Delta_n^3 }{6}\sum_{l< l^{\prime }}\sum_{k< k^{\prime }}\sum_{m< m^{\prime }}C_0^{Z,j_lj_{l^{\prime }}}C_0^{Z,j_kj_{k^{\prime }}}C_0^{Z,j_mj_{m^{\prime }}}\Big|\leq Kt^{7/2} \notag \\ &\mathbb{E}\Big(\sup_{w\in[0,s]}\Big\|Z_{t+w}-Z_t\Big\|^q \Big|\mathcal{F}_t \Big) \leq K_q s^{q/2},\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\Big\|\mathbb{E} \Big(Z_{t+s}-Z_t\Big)\Big|\mathcal{F}_t\Big\|\leq Ks. \\ \end{align} \end{lemma} \begin{lemma} Let ${\Greekmath 0110}_i^n$ be a $r$-dimensional $\mathcal{F}_i^n $ -measurable process satisfying $\|\mathbb{E}({\Greekmath 0110}_{i}^n|\mathcal{F} _{i-1}^n)\| \leq L^{\prime }$ and $\mathbb{E}\Big(\|{\Greekmath 0110}_{i}^n\|^q \Big| \mathcal{F}_{i-1}^n\Big) \leq L_q $. Also, let ${\Greekmath 0127}_i^n$ be a real-valued $\mathcal{F}_{i}^n$-measurable process with $\mathbb{E}\Big( \|{\Greekmath 0127}_{i+j-1}^n\|^q \Big|\mathcal{F}_{i-1}^n\Big) \leq L^q $ for $q \geq 2$ and $1\leq j \leq 2k_n-1$. Then, \begin{align*} \mathbb{E}\Bigg(\Bigg\|\sum_{j=1}^{2k_n-1}{\Greekmath 0127}_{i+j-1}^n{\Greekmath 0110}_{i+j}^n \Bigg\|^q \Bigg|\mathcal{F}_{i-1}^n\Bigg)\leq K_qL^q\Big(L_qk_n^{q/2}+L^{ \prime q}k_n^q\Big). \end{align*} \end{lemma} \begin{lemma} Under the assumptions of Theorem (ref), we have, for $i\in L\left( n,T\right) $: \begin{align*} & \Bigg|\mathbb{E}\Big(\left. {\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}{\Greekmath 0115} _{i+2k_{n}}^{n,gh}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}\right \vert \mathcal{F}_{i-1}^{n} \Big)-\frac{4}{k_{n}^{2}}\Big( C_{i-1}^{n,ga}C_{i-1}^{n,hb}+C_{i-1}^{n,gb}C_{i-1}^{n,ha})(C_{i-1}^{n,jl}C_{i-1}^{n,km}+C_{i-1}^{n,jm}C_{i-1}^{n,kl} \Big) \\ & -\frac{4\Delta _{n}}{3}\Big( C_{i-1}^{n,jl}C_{i-1}^{n,km}+C_{i-1}^{n,jm}C_{i-1}^{n,kl}\Big)\overline{C} _{i-1}^{n,gh,ab}-\frac{4\Delta _{n}}{3}\Big( C_{i-1}^{n,ga}C_{i-1}^{n,hb}-C_{i-1}^{n,gb}C_{i-1}^{n,ha}\Big)\overline{C} _{i-1}^{n,jk,lm} \\ & -\frac{4(k_{n}\Delta _{n})^{2}}{9}\overline{C}_{i-1}^{n,gh,ab}\overline{C} _{i-1}^{n,jk,lm}\Bigg|\leq K\Delta _{n}(\Delta _{n}^{1/8}+{\Greekmath 0111} _{i,4k_{n}}^{n} \Big). \end{align*} \end{lemma} \begin{lemma} Under the assumptions of Theorem (ref), we have, for $i\in L\left( n,T\right) $: \begin{align} & & \Big|\mathbb{E}\Big(\left. {\Greekmath 0117} _{i}^{n,jk}{\Greekmath 0117} _{i}^{n,lm}{\Greekmath 0117} _{i}^{n,gh}\right\vert \mathcal{F}_{i-1}^{n}\Big)\Big|& \leq K\Delta _{n}^{3/4} \Big(\Delta _{n}^{1/4}+{\Greekmath 0111} _{i,k_{n}}^{n}\Big), \\ & & \Big|\mathbb{E}\Big({\Greekmath 0117} _{i}^{n,jk}{\Greekmath 0117} _{i}^{n,lm}\left. \Big( C_{i+k_{n}-1}^{n,gh}-C_{i-1}^{n,gh}\Big)\right\vert \mathcal{F}_{i-1}^{n}\Big)\Big| & \leq K\Delta _{n}^{3/4}\Big(\Delta _{n}^{1/4}+{\Greekmath 0111} _{i,k_{n}}^{n}\Big), \\ & & \Big|\mathbb{E}\Big({\Greekmath 0117} _{i}^{n,jk}\Big(C_{i+k_{n}-1}^{n,lm}-C_{i-1}^{n,lm} \Big)\left. \Big(C_{i+k_{n}-1}^{n,gh}-C_{i-1}^{n,gh}\Big)\right\vert \mathcal{F} _{i-1}^{n}\Big)\Big|& \leq K\Delta _{n}^{3/4}\Big(\Delta _{n}^{1/4}+{\Greekmath 0111} _{i,k_{n}}^{n}\Big), \\ & & \Big|\mathbb{E}\Big(\left. {\Greekmath 0117} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}{\Greekmath 0115} _{i}^{n,gh}\right\vert \mathcal{F}_{i-1}^{n}\Big)\Big|& \leq K\Delta _{n}^{3/4} \Big(\Delta _{n}^{1/4}+{\Greekmath 0111} _{i,2k_{n}}^{n}\Big), \\ & & \Big|\mathbb{E}\Big(\left. {\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}{\Greekmath 0115} _{i}^{n,gh}\right\vert \mathcal{F}_{i-1}^{n}\Big)\Big|& \leq K\Delta _{n}^{3/4} \Big(\Delta _{n}^{1/4}+{\Greekmath 0111} _{i,2k_{n}}^{n}\Big). \end{align} \end{lemma} \begin{lemma} Under the assumptions of Theorem (ref), we have: \begin{align} & \frac{1}{\Delta _{n}^{1/4}}\sum_{i\in L\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{(i-2k_{n}-1)\Delta_n}){\Greekmath 011A} _{gh}(u,v)_{i}^{n}{\Greekmath 0110} _{ab}^{^{\prime }}(v)_{i}^{n}\overset{\mathbb{P}}{\Longrightarrow } 0, \forall (u,v) \\ & \frac{1}{\Delta _{n}^{1/4}}\Big(\overline{A11}(H,gh,u;G,ab,v)- \int_{0}^{T}(\partial _{gh}H\partial _{ab}G)(C_{t})\overline{C}_{t}^{gh,ab}dt \Big)\overset{\mathbb{P}}{\Longrightarrow }0\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{when} (u,v)=(2,2) \\ & \frac{1}{\Delta _{n}^{1/4}}\Big(\overline{A11}(H,gh,u;G,ab,v)-\frac{3}{ {\Greekmath 0112} ^{2}}\int_{0}^{T}(\partial _{gh}H\partial _{ab}G)(C_{t})(C_{t}^{ga}C_{t}^{hb}+C_{t}^{gb}C_{t}^{ha})dt\Big)\overset{ \mathbb{P}}{\Longrightarrow }0 \\ & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{when} (u,v)=(1,1), \notag \\ & \frac{1}{\Delta _{n}^{1/4}}\overline{A11}(H,gh,u;G,ab,v)\overset{\mathbb{P} }{\Longrightarrow }0\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{when} (u,v)=(1,2),(2,1) \end{align} \end{lemma} \section{Proof of Theorem (ref)} We now prove Theorem (ref). By Theorem (ref), we have \begin{align*} \frac{1}{\Delta _{n}^{1/4}}\Bigg(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A1\right) }& -\sum_{g,h,a,b=1}^{d}\sum_{u,v=1}^{2}\overline{A11} (H,gh,u;G,ab,v)_{T}^{n}+\overline{A12}(H,gh,u;G,ab,v)_{T}^{n} \\ & +\overline{A12}(G,ab,v;H,gh,u)_{T}^{n}\Bigg)\overset{\mathbb{P}}{ \Longrightarrow }0. \end{align*} Recalling the definition of $\overline{A12}(H,gh,u;G,ab,v)_{T}^{n}$ from equation ((ref)), Lemma (ref) implies that \begin{align} & \frac{1}{\Delta _{n}^{1/4}}\Bigg(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }-[H(C),G(C)]_{T}-\frac{3}{2k_{n}^{3}} \sum_{g,h,a,b}^{d}\sum_{u,v=1}^{2}\sum_{i\in L^{\prime }\left( n,T\right) } \\ & \Big[(\partial _{gh}H\partial _{ab}G)(C_{(i-2k_{n}-1)\Delta_n}){\Greekmath 011A} _{gh}(u,v)_{i}^{n}{\Greekmath 0110} _{ab}^{^{\prime \prime }}(v)_{i}^{n}+(\partial _{ab}H\partial _{gh}G)(C_{(i-2k_{n}-1)\Delta_n}){\Greekmath 011A} _{ab}(v,u)_{i}^{n}{\Greekmath 0110} _{gh}^{^{\prime \prime }}(v)_{i}^{n}\Big]\Bigg)\overset{\mathbb{P}}{ \Longrightarrow }0. \notag \end{align} Next, define \begin{align*} & {\Greekmath 0118} (H,gh,u;G,ab,v)_{i}^{n}=\frac{1}{\Delta _{n}^{1/4}}(\partial _{gh}H\partial _{ab}G)(C_{(i-2k_{n}-1)\Delta_n}){\Greekmath 011A} _{gh}(u,v)_{i}^{n}{\Greekmath 0110} _{ab}^{\prime \prime }(v)_{i}^{n}, \\ & Z(H,gh,u;G,ab,v)_{t}^{n}=\Delta _{n}^{1/4}\sum_{i=2k_{n}}^{[t/\Delta _{n}]}{\Greekmath 0118} (H,gh,u;G,ab,v)_{i}^{n}. \end{align*} Notice that ((ref)) implies \begin{align} & \frac{1}{\Delta _{n}^{1/4}}\Big(\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }-[H(C),G(C)]_{T}\Big)\overset{\mathcal{L}}{=} \sum_{g,h,a,b=1}^{d}\sum_{u,v=1}^{2}\frac{1}{\Delta _{n}^{1/4}}\Big( Z(H,gh,u;G,ab,v)_{T}^{n} \notag \\ & +Z(H,ab,v;G,gh,u)_{T}^{n}\Big). \end{align} The term ${\Greekmath 0123} _{i}^{\left( A\right) }$ depends on functions $H$ and $G$ , where we have so far suppressed the subscripts $r$, $r=1,..,{\Greekmath 0114} $, in the statement of Theorem (ref) for simplicity. Denote by ${\Greekmath 0123} _{i,r}^{\left( A\right) }$ the term ${\Greekmath 0123} _{i}^{\left( A\right) }$ that depends on functions $H_{r}$ and $G_{r}$. Observe that to derive the asymptotic distribution of $\left( \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i,1}^{\left( A\right) },...,\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i,{\Greekmath 0114} }^{\left( A\right) }\right) $ , it suffices to study the joint asymptotic behavior of the family of processes $\frac{1}{\Delta _{n}^{1/4}} Z(H,gh,u;G,ab,v)_{T}^{n}$. Notice that ${\Greekmath 0118} (H,gh,u;G,ab,v)_{i}^{n}$ are martingale increments relative to the discrete filtration $(\mathcal{F} _{i}^{n})$. Therefore, to obtain the joint asymptotic distribution of $\frac{ 1}{\Delta _{n}^{1/4}}Z(H,gh,u;G,ab,v)_{T}^{n}$, it is enough to prove the following three properties: \begin{align} & A\Big((H,gh,u;G,ab,v),(H^{\prime },g^{\prime }h^{\prime },u^{\prime };G^{\prime },a^{\prime }b^{\prime },v^{\prime })\Big)_{t}^{n} \notag \\ & =\sum_{i\in L^{\prime }\left( n,T\right) }\mathbb{E}({\Greekmath 0118} (H,gh,u;G,ab,v)_{i}^{n}{\Greekmath 0118} (H^{\prime },g^{\prime }h^{\prime },u^{\prime };G^{\prime },a^{\prime }b^{\prime },v^{\prime })_{i}^{n}|\mathcal{F} _{i-1}^{n}) \\ & \overset{\mathbb{P}}{\Longrightarrow }A\Big( (H,gh,u;G,ab,v),(H^{\prime },g^{\prime }h^{\prime },u^{\prime };G^{\prime },a^{\prime }b^{\prime },v^{\prime })\Big)_{t}, \\ & \sum_{i\in L^{\prime }\left( n,T\right) }\mathbb{E}\Big(\Big|{\Greekmath 0118} (H,gh,u;G,ab,v\Big)_{i}^{n}\Big|^{4}\Big|\mathcal{F}_{i-1}^{n})\overset{ \mathbb{P}}{\Longrightarrow }0,\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and} \\ & B(N;H,gh,u;G,ab,v)_{t}^{n}:=\sum_{i\in L^{\prime }\left( n,T\right) } \mathbb{E}\Big({\Greekmath 0118} (H,gh,u;G,ab,v)_{i}^{n}\Delta _{i}^{n}N|\mathcal{F} _{i-1}^{n}\Big)\overset{\mathbb{P}}{\Longrightarrow }0, \end{align} for all $t>0$, all $(H,gh,u;G,ab,v),(H^{\prime },g^{\prime }h^{\prime },u^{\prime };G^{\prime },a^{\prime }b^{\prime },v^{\prime })$ and all martingales $N$ which are either bounded and orthogonal to $W$, or equal to one component $W^{j}$.\newline Since the derivatives of $H_{r}$ and $G_{r}$ are bounded, equations ((ref)) and ((ref)) can be proved by an extension of (B.105) and (B.106) in yacjacod14 to multivariate processes.\newline Next, define \begin{equation*} V_{ab}^{a^{\prime }b^{\prime }}(v,v^{\prime })_{t}= \begin{cases} (C_{t}^{aa^{\prime }}C_{t}^{bb^{\prime }}+C_{t}^{ab^{\prime }}C_{t}^{ba^{\prime }}) & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(v,v^{\prime })=(1,1) \\ \overline{C}_{t}^{ab,a^{\prime }b^{\prime }} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} (v,v^{\prime })=(2,2) \\ 0 & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}. \end{cases} \end{equation*} Using again the boundedness of the derivatives of $H_{r}$ and $G_{r}$, we can show that \begin{align*} A\Big((H,gh,u;G,ab,v),& (H^{\prime },g^{\prime }h^{\prime },u^{\prime };G^{\prime },a^{\prime }b^{\prime },v^{\prime })\Big)_{t}= \\ & M(u,v;u^{\prime },v^{\prime })\int_{0}^{t}(\partial _{gh}H\partial _{ab}G\partial _{g^{\prime }h^{\prime }}H\partial _{a^{\prime }b^{\prime }}G)(C_{s})V_{ab}^{a^{\prime }b^{\prime }}(v,v^{\prime })_{s}V _{gh}^{g^{\prime }h^{\prime }}(u,u^{\prime })_{s}ds, \end{align*} with \begin{equation*} M(u,v;u^{\prime },v^{\prime })= \begin{cases} 3/{\Greekmath 0112} ^{3} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v;u^{\prime },v^{\prime })=(1,1;1,1) \\ 3/4{\Greekmath 0112} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v;u^{\prime },v^{\prime })=(1,2;1,2),(2,1;2,1) \\ 151{\Greekmath 0112} /280 & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v;u^{\prime },v^{\prime })=(2,2;2,2) \\ 0 & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}. \end{cases} \end{equation*} Therefore, we have \newline $A\Big((H,gh,u;G,ab,v),(H^{\prime },g^{\prime }h^{\prime },u^{\prime };G^{\prime },a^{\prime }b^{\prime },v^{\prime })\Big)_{T}$ = \begin{equation*} \begin{cases} \frac{3}{{\Greekmath 0112} ^{3}}\int_{0}^{T}(\partial _{gh}H\partial _{ab}G\partial _{g^{\prime }h^{\prime }}H^{\prime }\partial _{a^{\prime }b^{\prime }}G^{\prime })(C_{t})(C_{t}^{gg^{\prime }}C_{t}^{hh^{\prime }}+C_{t}^{gh^{\prime }}C_{t}^{hg^{\prime }})(C_{t}^{aa^{\prime }}C_{t}^{bb^{\prime }}+C_{t}^{ab^{\prime }}C_{t}^{ba^{\prime }})dt, \\ \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v;u^{\prime },v^{\prime })=(1,1;1,1) \\ \frac{3}{4{\Greekmath 0112} }\int_{0}^{T}(\partial _{gh}H\partial _{ab}G\partial _{g^{\prime }h^{\prime }}H^{\prime }\partial _{a^{\prime }b^{\prime }}G^{\prime })(C_{t})(C_{t}^{gg^{\prime }}C_{t}^{hh^{\prime }}+C_{t}^{gh^{\prime }}C_{t}^{hg^{\prime }})\overline{C}_{t}^{ab,a^{\prime }b^{\prime }}dt, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v;u^{\prime },v^{\prime })=(1,2;1,2) \\ \frac{3}{4{\Greekmath 0112} }\int_{0}^{T}(\partial _{gh}H\partial _{ab}G\partial _{g^{\prime }h^{\prime }}H^{\prime }\partial _{a^{\prime }b^{\prime }}G^{\prime })(C_{t})(C_{t}^{aa^{\prime }}C_{t}^{bb^{\prime }}+C_{t}^{ab^{\prime }}C_{s}^{ba^{\prime }})\overline{C}_{t}^{gh,g^{\prime }h^{\prime }}dt, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v;u^{\prime },v^{\prime })=(2,1;2,1) \\ \frac{151{\Greekmath 0112} }{280}\int_{0}^{T}(\partial _{gh}H\partial _{ab}G\partial _{g^{\prime }h^{\prime }}H^{\prime }\partial _{a^{\prime }b^{\prime }}G^{\prime })(C_{t})\overline{C}_{s}^{ab,a^{\prime }b^{\prime }}\overline{C} _{t}^{gh,g^{\prime }h^{\prime }}dt, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} (u,v;u^{\prime },v^{\prime })=(2,2;2,2) \\ 0 \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}. \end{cases} \end{equation*} Using equation ((ref)), we deduce that the asymptotic covariance between $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i,r}^{\left( A\right) }$ and $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i,s}^{\left( A\right) }$ is given by \begin{eqnarray*} &&\sum_{g,h,a,b=1}^{d}\sum_{g^{\prime },h^{\prime },a^{\prime },b^{\prime }=1}^{d}\sum_{u,v,u^{\prime },v^{\prime }=1}^{2}\Bigg(A\Big( (H_{r},gh,u;G_{r},ab,v),(H_{s},g^{\prime }h^{\prime },u^{\prime };G_{s},a^{\prime }b^{\prime },v^{\prime })\Big)_{T} \\ &&+A\Big((H_{r},gh,u;G_{r},ab,v),(H_{s},a^{\prime }b^{\prime },v^{\prime };G_{s},g^{\prime }h^{\prime },u^{\prime })\Big)_{T} \\ &&+A\Big((H_{r},ab,v;G_{r},gh,u),(H_{s},g^{\prime }h^{\prime },u^{\prime };G_{s},a^{\prime }b^{\prime },v^{\prime })\Big)_{T} \\ &&+A\Big((H_{r},ab,v;H_{r},gh,u),(H_{s},a^{\prime }b^{\prime },v^{\prime };G_{s},g^{\prime }h^{\prime },u^{\prime })\Big)_{T}\Bigg). \end{eqnarray*} The above expression can be rewritten as \begin{eqnarray*} &&\sum_{g,h,a,b=1}^{d}\sum_{j,k,l,m=1}^{d}\Bigg(\frac{6}{{\Greekmath 0112} ^{3}} \int_{0}^{T}\big(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s}(C_{t})\big)\Big[ (C_{t}^{gj}C_{t}^{hk}+C_{t}^{gk}C_{t}^{hj})(C_{t}^{al}C_{t}^{bm}+C_{t}^{am}C_{t}^{bl}) \\ &&+(C_{t}^{aj}C_{t}^{bk}+C_{t}^{ak}C_{t}^{bj})(C_{t}^{gl}C_{t}^{hm}+C_{t}^{gm}C_{t}^{hl}) \Big]dt \\ &&+\frac{151{\Greekmath 0112} }{140}\int_{0}^{t}\big(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s}(C_{t})\big)\Big[\overline{C} _{t}^{gh,jk}\overline{C}_{t}^{ab,lm}+\overline{C}_{t}^{ab,jk}\overline{C} _{t}^{gh,lm}\Big]dt \\ &&+\frac{3}{2{\Greekmath 0112} }\int_{0}^{t}\big(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s}(C_{t})\big)\Big[ (C_{t}^{gj}C_{t}^{hk}+C_{t}^{gk}C_{t}^{hj})\overline{C} _{t}^{ab,lm}+(C_{t}^{al}C_{t}^{bm}+C_{t}^{am}C_{t}^{bl})\overline{C} _{t}^{gh,jk} \\ &&+(C_{t}^{gl}C_{s}^{hm}+C_{t}^{gm}C_{s}^{hl})\overline{C} _{t}^{ab,jk}+(C_{t}^{aj}C_{t}^{bk}+C_{t}^{ak}C_{t}^{bj})\overline{C} _{t}^{gh,lm}\Big]dt\Bigg), \end{eqnarray*} which completes the proof. \section{Proof of Theorem (ref)} Recall that $N_{s}$ is the number of jumps in $C$ from time $0$ to $s$. Let \begin{eqnarray*} L^{\prime \prime }\left( n\right) &=&\left\{ i=k_{n}+1,k_{n}+2,...:N_{\left( i+5\right) k_{n}\Delta _{n}}-N_{\left( i-1\right) k_{n}\Delta _{n}}=0\right\} , \\ L^{\prime \prime }\left( n,T\right) &=&\left\{ i=1,2,...,\left[ T/\Delta _{n} \right] -5k_{n}+1\right\} \cap L^{\prime \prime }\left( n\right) , \\ \overline{L}^{\prime \prime }\left( n,T\right) &=&\left\{ i=1,2,...,\left[ T/\Delta _{n}\right] -5k_{n}+1\right\} \backslash L^{\prime \prime }\left( n\right) . \end{eqnarray*} Denote by $\widehat{{\Greekmath 0121} }_{T}^{r,s,(1)}$, $\widehat{{\Greekmath 0121} } _{T}^{r,s,(2)} $, and $\widehat{{\Greekmath 0121} }_{T}^{r,s,(3)}$ the $i^{th}$ summand of $\widehat{\Omega }_{T}^{r,s,(1)}$, $\widehat{\Omega }_{T}^{r,s,(2)}$, and $\widehat{\Omega }_{T}^{r,s,(3)}$, without the volatility jump truncation, so they satisfy \begin{equation*} \widehat{\Omega }_{T}^{r,s,(m)}=\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-5k_{n}+1} \widehat{{\Greekmath 0121} }_{T}^{r,s,(m)}1_{\left\{ A_{i}\cap A_{i+k_{n}}\cap A_{i+2k_{n}}\cap A_{i+3k_{n}}\right\} }\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }m=1,2\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }3\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{. } \end{equation*} The same methods as in Theorems (ref) and (ref) can be used to show \begin{eqnarray*} \sum_{i\in L^{\prime \prime }\left( n,T\right) }\widehat{{\Greekmath 0121} } _{T}^{r,s,(m)}1_{\left\{ A_{i}\cap A_{i+k_{n}}\cap A_{i+2k_{n}}\cap A_{i+3k_{n}}\right\} }-\sum_{i\in L^{\prime \prime }\left( n,T\right) } \widehat{{\Greekmath 0121} }_{T}^{r,s,(m)} &=&o_{p}\left( 1\right) \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and} \\ \sum_{i\in \overline{L}^{\prime \prime }\left( n,T\right) }\widehat{{\Greekmath 0121} } _{T}^{r,s,(m)}1_{\left\{ A_{i}\cap A_{i+k_{n}}\cap A_{i+2k_{n}}\cap A_{i+3k_{n}}\right\} } &=&o_{p}\left( 1\right) . \end{eqnarray*} We conclude that the probability limit of $\widehat{\Omega }_{T}^{r,s,(m)}$ is the same as $\sum_{i\in L^{\prime \prime }\left( n,T\right) }\widehat{ {\Greekmath 0121} }_{T}^{r,s,(m)}$ for $m=1,2,3$. Using boundedness of the derivatives of $H_{r},G_{r},H_{s}$ and $G_{s}$ and Theorem 2.2 in jacodrosenbaum-sqrtn, one can show that \begin{equation*} \frac{6}{{\Greekmath 0112} ^{3}}\sum_{i\in L^{\prime \prime }\left( n,T\right) } \widehat{{\Greekmath 0121} }_{T}^{r,s,(1)}\overset{\mathbb{P}}{\longrightarrow }\Sigma _{T}^{r,s,(1)}. \end{equation*} Next, by equation (3.27) in jacodrosenbaum-sqrtn, we have \begin{equation*} \frac{3}{2{\Greekmath 0112} }\left( \sum_{i\in L^{\prime \prime }\left( n,T\right) } \widehat{{\Greekmath 0121} }_{T}^{r,s,(3)}-\frac{6}{{\Greekmath 0112} }\sum_{i\in L^{\prime \prime }\left( n,T\right) }\widehat{{\Greekmath 0121} }_{T}^{r,s,(1)}\right) \overset{\mathbb{P }}{\longrightarrow }\Sigma _{T}^{r,s,(3)}. \end{equation*} Finally, to show that \begin{equation*} \frac{151{\Greekmath 0112} }{140}\frac{9}{4{\Greekmath 0112} ^{2}}\left( \sum_{i\in L^{\prime \prime }\left( n,T\right) }\widehat{{\Greekmath 0121} }_{T}^{r,s,(2)}+\frac{4}{{\Greekmath 0112} ^{2}}\sum_{i\in L^{\prime \prime }\left( n,T\right) }\widehat{{\Greekmath 0121} } _{T}^{r,s,(1)}-\frac{4}{3}\sum_{i\in L^{\prime \prime }\left( n,T\right) } \widehat{{\Greekmath 0121} }_{T}^{r,s,(3)}\right) \overset{\mathbb{P}}{\longrightarrow } \Sigma _{T}^{r,s,(2)}, \end{equation*} we first observe that the approximation error induced by replacing $\widehat{ C}_{i}^{n}$ by $\widehat{C}_{i}^{^{\prime }n}$ in Theorem (ref) is negligible.\newline For $1\leq g,h,a,b,j,k,l,m\leq d$ and $1\leq r,s\leq d$, we define \begin{align*} \widehat{W}_{T}^{n}& =\sum_{i\in L^{\prime \prime }\left( n,T\right) }(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{gh}H_{s}\partial _{lm}G_{s})(\widehat{C}_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}{\Greekmath 0115} _{i+2k_{n}}^{n,lm}, \\ \widehat{w}(1)_{i}^{n}& =(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(C_{i-1}^{n})\mathbb{E}({\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}{\Greekmath 0115} _{i+2k_{n}}^{n,lm}|\mathcal{F}_{i}^{n}), \\ \widehat{w}(2)_{i}^{n}& =(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(C_{i-1}^{n})({\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}{\Greekmath 0115} _{i+2k_{n}}^{n,lm}-\mathbb{E} ({\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}{\Greekmath 0115} _{i+2k_{n}}^{n,lm}|\mathcal{F}_{i}^{n})), \\ \widehat{w}(3)_{i}^{n}& =\Big((\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(\widehat{C} _{i}^{n})-(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(C_{i-1}^{n})\Big){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}{\Greekmath 0115} _{i+2k_{n}}^{n,lm}, \\ \widehat{W}(u)_{t}^{n}& =\sum_{i\in L^{\prime \prime }\left( n,T\right) } \widehat{w}_{i}(u),u=1,2,3. \end{align*} Now, note that we also have $\widehat{W}_{t}^{n}=\widehat{W}(1)_{t}^{n}+ \widehat{W}(2)_{t}^{n}+\widehat{W}(3)_{t}^{n}$. By Taylor expansion and using repeatedly the boundedness of $C_{t}$, we obtain, for $i\in L^{\prime \prime }\left( n,T\right) $ \begin{equation*} |\widehat{w}(3)_{i}^{n}|\leq K\Vert {\Greekmath 0117} _{i}^{n}\Vert \Vert {\Greekmath 0115} _{i}^{n}\Vert ^{2}\Vert {\Greekmath 0115} _{i+2k_{n}}^{n}\Vert ^{2}, \end{equation*} which implies $\mathbb{E}(|\widehat{w}(3)_{i}^{n}|)\leq K\Delta _{n}^{5/4}$ and hence $\widehat{W}(3)_{t}^{n}\overset{\mathbb{P}}{\longrightarrow }0$. Using Cauchy-Schwartz inequality and the bound $\mathbb{E}(\Vert {\Greekmath 0115} _{i}^{n}\Vert ^{q}|\mathcal{F}_{i}^{n})\leq K\Delta _{n}^{q/4}$, we have $ \mathbb{E}(|\widehat{w}(2)_{i}^{n}|^{2})\leq K\Delta _{n}^{2}$ for $i\in L^{\prime \prime }\left( n,T\right) $. Observing furthermore that $\widehat{w }(2)_{i}^{n}$ is $\mathcal{F}_{i+4k_{n}}-$measurable, Lemma B.8 in yacjacod14 implies $\widehat{W}(2)_{t}^{n}\overset{\mathbb{P}}{ \longrightarrow }0$. \newline Next, define \begin{align*} & w_{i}^{n}=(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(C_{i-1}^{n})\Big[\frac{4}{k_{n}^{2}\Delta _{n}} (C_{i-1}^{n,ga}C_{i-1}^{n,hb}+C_{i-1}^{n,gb}C_{i-1}^{n,ha})(C_{i-1}^{n,jl}C_{i-1}^{n,km}+C_{i-1}^{n,jm}C_{i-1}^{n,kl}) \\ & +\frac{4}{3}(C_{i-1}^{n,jl}C_{i-1}^{n,km}+C_{i-1}^{n,jm}C_{i-1}^{n,kl}) \overline{C}_{i-1}^{n,gh,ab}+\frac{4}{3} (C_{i-1}^{n,ga}C_{i-1}^{n,hb}+C_{i-1}^{n,gb}C_{i-1}^{n,ha})\overline{C}_{i-1}^{n,jk,lm} \\ & +\frac{4(k_{n}^{2}\Delta _{n})}{9}\overline{C}_{i-1}^{n,gh,ab} \overline{C}_{i-1}^{n,jk,lm}\Big], \\ & W_{T}^{n}=\Delta _{n}\sum_{i\in L^{\prime \prime }\left( n,T\right) }w_{i}^{n}. \end{align*} Using the cadlag property of $c$ and $\overline{C}$, $k_{n}\sqrt{\Delta _{n}} \rightarrow {\Greekmath 0112} $, and the Riemann integral convergence, we conclude that $W_{T}^{n}\overset{\mathbb{P}}{\longrightarrow }W_{T}$ where \begin{align*} & W_{T}=\int_{0}^{T}(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(C_{t})\Big[\frac{4}{{\Greekmath 0112} ^{2}} (C_{t}^{ga}C_{t}^{hb}+C_{t}^{gb}C_{t}^{ha})(C_{t}^{jl}C_{t}^{km}+C_{t}^{jm}C_{t}^{kl}) \\ & +\frac{4}{3}(C_{t}^{jl}C_{t}^{km}+C_{t}^{jm}C_{t}^{kl})\overline{C} _{t}^{gh,ab}+\frac{4}{3}(C_{t}^{ga}C_{i}^{hb}+C_{t}^{gb}C_{t}^{ha})\overline{ C}_{t}^{jk,lm}+\frac{4{\Greekmath 0112} ^{2}}{9}\overline{C}_{t}^{gh,ab}\overline{C} _{t}^{jk,lm}\Big]dt. \end{align*} In addition, by Lemma (ref), it holds that \begin{equation*} \mathbb{E}(|\widehat{W}(1)_{T}^{n}-W_{T}^{n}|)\leq \Delta _{n}\mathbb{E} \Bigg(\sum_{i\in L^{\prime \prime }\left( n,T\right) }(\Delta _{n}^{1/8}+{\Greekmath 0111} _{i,4k_{n}})\Bigg). \end{equation*} Hence, by the third result of Lemma (ref) we have $\widehat{W} _{T}^{n}\overset{\mathbb{P}}{\longrightarrow }W_{t}$, from which it follows that \begin{align*} & \frac{9}{4{\Greekmath 0112} ^{2}}\Big[\widehat{W}(1)_{T}^{n}+\frac{4}{k_{n}^{2}} \sum_{i\in L^{\prime \prime }\left( n,T\right) }(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(\widehat{C} _{i}^{n})[C_{i}^{n}(jk,lm)C_{i}^{n}(gh,ab)] \\ & -\frac{2}{k_{n}}\sum_{i\in L^{\prime \prime }\left( n,T\right) }(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})( \widehat{C}_{i}^{n})C_{i}^{n}(gh,ab){\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm} \\ & -\frac{2}{k_{n}}\sum_{i\in L^{\prime \prime }\left( n,T\right) }(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})( \widehat{C}_{i}^{n})C_{i}^{n}(jk,lm){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab} \Big] \\ & \overset{\mathbb{P}}{\longrightarrow }\int_{0}^{T}(\partial _{gh}H_{r}\partial _{ab}G_{r}\partial _{jk}H_{s}\partial _{lm}G_{s})(C_{t}) \overline{C}_{t}^{gh,ab}\overline{C}_{t}^{jk,lm}dt. \end{align*} The result follows from the above convergence, the already invoked symmetry argument, and straightforward calculations. \section{Proofs of Auxiliary Lemmas and Theorems} This section is devoted to the proofs of the auxiliary theorems and lemmas (listed in Section (ref)) that were used to prove Theorem (ref) and Theorem (ref). \subsection{Proof of Theorem (ref)} The proof proceeds in three steps. In Step 1, we prove, for $i\in L\left( n,T\right) $, \begin{equation} P\left( \overline{A}_{i}\right) \leq Ka_{n}\Delta _{n}^{\left( 2-r\right) {\Greekmath 0124} -{\Greekmath 0124} ^{\prime }}, \end{equation} where $a_{n}$ is a sequence converging to zero, and $\overline{A}_{i}$ is the complement of $A_{i}$. In Step 2, we prove, for $p\geq 1$ and $i\in L\left( n,T\right) $, \begin{equation} E\left[ \left\vert {\Greekmath 0123} _{i}\right\vert ^{p}\right] \leq K\Delta _{n}^{p}+Ka_{n}\Delta _{n}^{\left( 4p-r\right) {\Greekmath 0124} +1-\frac{3}{2}p}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{. } \end{equation} Step 3 completes the proof of Theorem (ref). Step 1. We now prove equation ((ref)). Recall $\widehat{C}_{i}^{\prime n}$ notation in ((ref)). For $i\in L\left( n,T\right) $, \begin{eqnarray} P\left( \overline{A}_{i}\right) &=&P\left( \left\Vert \widehat{C} _{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}\right\Vert \geq u_{n}^{\prime }\right) \notag \\ &\leq &P\left( \left\Vert \widehat{C}_{i+k_{n}}^{\prime n}-\widehat{C} _{i-k_{n}}^{\prime n}\right\Vert +\left\Vert \widehat{C}_{i+k_{n}}^{n}- \widehat{C}_{i+k_{n}}^{\prime n}\right\Vert +\left\Vert \widehat{C} _{i-k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{\prime n}\right\Vert \geq u_{n}^{\prime }\right) \notag \\ &\leq &P\left( \left\Vert \widehat{C}_{i+k_{n}}^{\prime n}-\widehat{C} _{i-k_{n}}^{\prime n}\right\Vert \geq \frac{u_{n}^{\prime }}{2}\right) +P\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i+k_{n}}^{\prime n}\right\Vert +\left\Vert \widehat{C}_{i-k_{n}}^{n}-\widehat{C} _{i-k_{n}}^{\prime n}\right\Vert \geq \frac{u_{n}^{\prime }}{2}\right) . \end{eqnarray} Using standard results in the literature, we have for $q\geq 2$ and $i\in L\left( n,T\right) $, \begin{equation} E\left( \left\Vert \widehat{C}_{i+k_{n}}^{\prime n}-\widehat{C} _{i-k_{n}}^{\prime n}\right\Vert ^{q}\right) \leq K\Delta _{n}^{q/4}, \end{equation} see, for example, equation (3.26) in jacodrosenbaum-sqrtn. Therefore, the first term in ((ref)) satisfies, by Markov's inequality, for $p\geq 2$, \begin{equation} P\left( \left\Vert \widehat{C}_{i+k_{n}}^{\prime n}-\widehat{C} _{i-k_{n}}^{\prime n}\right\Vert \geq \frac{u_{n}^{\prime }}{2}\right) \leq K\Delta _{n}^{p/4-{\Greekmath 0124} ^{\prime }p}. \end{equation} By (4.8) in jacodrosenbaum13, there exists a sequence of real numbers $ a_{n} $ converging to zero such that \begin{equation} \mathbb{E}(\Vert \widehat{C}_{i}^{n}-\widehat{C}_{i}^{^{\prime }n}\Vert ^{q})\leq K_{q}a_{n}\Delta _{n}^{(2q-r){\Greekmath 0124} +1-q}, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for any}q\geq 1, \end{equation} where for later use, we note that this result also holds in the presence of volatility jumps. Therefore, the second term in ((ref)) satisfies, by Markov's inequality, \begin{eqnarray} &&P\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i+k_{n}}^{\prime n}\right\Vert +\left\Vert \widehat{C}_{i-k_{n}}^{n}-\widehat{C} _{i-k_{n}}^{\prime n}\right\Vert \geq \frac{u_{n}^{\prime }}{2}\right) \notag \\ &\leq &\frac{1}{u_{n}^{\prime }/2}E\left( \left\Vert \widehat{C} _{i+k_{n}}^{n}-\widehat{C}_{i+k_{n}}^{\prime n}\right\Vert +\left\Vert \widehat{C}_{i-k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{\prime n}\right\Vert \right) \leq Ka_{n}\Delta _{n}^{\left( 2-r\right) {\Greekmath 0124} -{\Greekmath 0124} ^{\prime }}. \end{eqnarray} Since ${\Greekmath 0124} ^{\prime }<\frac{1}{8}$ and by choosing sufficiently large $p$ in ((ref)), equations ((ref)) and ((ref)) give ((ref)). Step 2. We now prove equation ((ref)). First, note that for $q\geq 1$, by ((ref)), \begin{equation} \mathrm{E}\left( \left\Vert \widehat{C}_{i\Delta _{n}}^{n}\right\Vert ^{q}\right) \leq K\mathrm{E}\left[ \left\vert \widehat{C}_{i\Delta _{n}}^{n}- \widehat{C}_{i\Delta _{n}}^{\prime n}\right\vert ^{q}\right] +K\mathrm{E} \left[ \left\vert \widehat{C}_{i\Delta _{n}}^{\prime n}\right\vert ^{q} \right] \leq Ka_{n}\Delta _{n}^{\left( 2q-r\right) {\Greekmath 0124} +1-q}+K. \end{equation} By Taylor expansion and $H$ and $G$ having bounded derivatives, for $ i\in L\left( n,T\right) $ and $p\geq 1$, \begin{eqnarray} &&E\left[ \left\vert {\Greekmath 0123} _{i}\right\vert ^{p}\right] \notag \\ &\leq & K\frac{1}{ k_{n}^{p}}\mathrm{E}\left( \left\Vert \widehat{C}_{\left( i+k_{n}\right) \Delta _{n}}^{n}-\widehat{C}_{i\Delta _{n}}^{n}\right\Vert ^{2p}\right) +K \frac{1}{k_{n}^{2p}}\mathrm{E}\left( \left\Vert \widehat{C}_{i\Delta _{n}}^{n}\right\Vert ^{2p}\right) \notag \\ &\leq & K\Delta _{n}^{p/2}\mathrm{E}\left( \left\Vert \widehat{C}_{\left( i+k_{n}\right) \Delta _{n}}^{\prime n}-\widehat{C}_{i\Delta _{n}}^{\prime n}\right\Vert ^{2p}+\left\Vert \widehat{C}_{\left( i+k_{n}\right) \Delta _{n}}^{n}- \widehat{C}_{\left( i+k_{n}\right) \Delta _{n}}^{\prime n}\right\Vert ^{2p}+\left\Vert \widehat{C}_{i\Delta _{n}}^{n}-\widehat{C}_{i\Delta _{n}}^{n\prime }\right\Vert ^{2p}\right) +K\Delta _{n}^{p}\mathrm{E}\left( \left\Vert \widehat{C}_{i\Delta _{n}}^{n}\right\Vert ^{2p}\right) \notag \\ &\leq &K\Delta _{n}^{p/2}\left( \Delta _{n}^{p/2}+a_{n}\Delta _{n}^{\left( 4p-r\right) {\Greekmath 0124} +1-2p}\right) +K\Delta _{n}^{p}\left[ Ka_{n}\Delta _{n}^{\left( 4p-r\right) {\Greekmath 0124} +1-2p}+K\right] \notag \\ &=&K\Delta _{n}^{p}+Ka_{n}\Delta _{n}^{\left( 4p-r\right) {\Greekmath 0124} +1-\frac{3}{ 2}p}, \end{eqnarray} where the third inequality uses ((ref)), ((ref)) and ((ref)). Step 3. We now complete the proof of Theorem (ref). By the triangle and Cauchy-Schwarz inequalities, \begin{eqnarray*} &&E\left\vert \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}\right\vert \\ &\leq &\sum_{i\in L\left( n,T\right) }E\left\vert {\Greekmath 0123} _{i}\left( 1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }-1\right) \right\vert \\ &\leq &\sum_{i\in L\left( n,T\right) }\sqrt{E\left\vert {\Greekmath 0123} _{i}^{2}\right\vert }\sqrt{P\left( \overline{A_{i}}\cup \overline{A_{i+k_{n}} }\right) } \\ &\leq &\sum_{i\in L\left( n,T\right) }\sqrt{E\left\vert {\Greekmath 0123} _{i}^{2}\right\vert }\sqrt{P\left( \overline{A_{i}}\right) +P\left( \overline{A_{i+k_{n}}}\right) } \\ &\leq &K\Delta _{n}^{-1}\left( \Delta _{n}^{\left( 8-r\right) {\Greekmath 0124} -2}\right) ^{1/2}\left( a_{n}\Delta _{n}^{\left( 2-r\right) {\Greekmath 0124} -{\Greekmath 0124} ^{\prime }}\right) ^{1/2} \\ &=&\Delta _{n}^{l\left( {\Greekmath 0124} ,{\Greekmath 0124} ^{\prime }\right) }, \end{eqnarray*} where 4th inequality follows by ((ref)) with $p=2$, and ((ref)). In the above, \begin{equation*} l\left( {\Greekmath 0124} ,{\Greekmath 0124} ^{\prime }\right) =-1+\frac{1}{2}\left[ \left( 8-r\right) {\Greekmath 0124} -2\right] +\frac{1}{2}\left[ \left( 2-r\right) {\Greekmath 0124} -{\Greekmath 0124} ^{\prime }\right] . \end{equation*} A straightforward calculation shows that ${\Greekmath 0124} > \frac{2{\Greekmath 0124} ^{\prime }+9}{4\left( 5-r\right) }$ implies $l\left( {\Greekmath 0124} ,{\Greekmath 0124} ^{\prime }\right) > \frac{1}{4}$, which completes the proof of Theorem (ref). \subsection{Proof of Theorem (ref)} Without loss of generality, we can assume that there is at most one volatility jump in $\left( \left( i-k_{n}\right) \Delta _{n},\left( i+3k_{n}\right) \Delta _{n}\right] $ for any $i\in \overline{L}\left( n,T\right) $. To study the behavior of ${\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }$ on $i\in \overline{L}\left( n,T\right) $, we will distinguish between two cases, depending on whether or not there is a volatility jump in $\left( i\Delta _{n},\left( i+2k_{n}\right) \Delta _{n} \right] $. So define $B_{i}$ as the event that there is a volatility jump in $\left( i\Delta _{n},\left( i+2k_{n}\right) \Delta _{n}\right] $ (we omit indexing $B_{i}$\ by $n$ for brevity). Denote by $\overline{B}_{i}$ the complement of $B_{i}$. Intuitively, for $i\in \overline{L}\left( n,T\right) $ , ${\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }$ is small because, on the one hand, $P\left( A_{i}\cap A_{i+k_{n}}\right) $ is small on $B_{i}$, on the other hand, ${\Greekmath 0123} _{i}$ is small on $\overline{B} _{i} $. We have \begin{eqnarray} &&\mathrm{E}\left\vert \sum_{i\in \overline{L}\left( n,T\right) }{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }\right\vert =\mathrm{E} \left\vert \sum_{i\in \overline{L}\left( n,T\right) :B_{i}}{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }+\sum_{i\in \overline{L}\left( n,T\right) :\overline{B}_{i}}{\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }\right\vert \notag \\ &\leq &\sum_{i\in \overline{L}\left( n,T\right) :B_{i}}\mathrm{E}\left\vert {\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }\right\vert +\sum_{i\in \overline{L}\left( n,T\right) :\overline{B}_{i}}\mathrm{E} \left\vert {\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }\right\vert , \end{eqnarray} where \textquotedblleft $i\in \overline{L}\left( n,T\right) :B_{i}$ \textquotedblright\ denotes those terms in $\overline{L}\left( n,T\right) $, for which $B_{i}$ is true. First, we show that the second term in ((ref)) is $ o_{p}\left( \Delta _{n}^{1/4}\right) $. For $i\in \overline{L}\left( n,T\right) $ such that $B_{i}$ if false, we can use the bound on $\mathrm{E} \left[ \left\vert {\Greekmath 0123} _{i}\right\vert ^{p}\right] $ in ((ref)) for $p\geq 1$. The second term in ( (ref)) satisfies \begin{eqnarray*} &&\sum_{i\in \overline{L}\left( n,T\right) :\overline{B}_{i}}\mathrm{E} \left\vert {\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }\right\vert \\ &\leq &\sum_{i\in \overline{L}\left( n,T\right) :\overline{B}_{i}}\mathrm{E} \left\vert {\Greekmath 0123} _{i}\right\vert \\ &\leq &K\Delta _{n}^{-1/2}\left( \Delta _{n}+\Delta _{n}^{\left( 4-r\right) {\Greekmath 0124} -\frac{1}{2}}\right) \\ &=&K\Delta _{n}^{1/2}+K\Delta _{n}^{\left( 4-r\right) {\Greekmath 0124} -1}. \end{eqnarray*} Theorem 1 assumptions imply $\left( 4-r\right) {\Greekmath 0124} -1> \frac{1}{4}$, so the second term in ((ref)) is $o_{p}\left( \Delta _{n}^{1/4}\right) $. The rest of the proof is devoted to showing that the first term in ((ref)) is $o_{p}\left( \Delta _{n}^{1/4}\right) $. This will complete the proof of Theorem (ref). The first term in ((ref)) involves those $i\in \overline{L} \left( n,T\right) $, for which $B_{i}$ is true. We will show below that \begin{equation} P\left( A_{i}\cap A_{i+k_{n}}\right) \leq K\Delta _{n}^{1/2}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i\in \overline{L}\left( n,T\right) \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ such that }B_{i}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ holds.} \end{equation} We use the following bound in the presence of the volatility jump, \begin{equation} \mathrm{E}\left( \left\vert {\Greekmath 0123} _{i}\right\vert ^{p}\right) \leq K \frac{1}{k_{n}^{p}}\left[ \mathrm{E}\left[ \left\vert \widehat{C} _{i}^{n}\right\vert ^{p}\right] +\mathrm{E}\left[ \left\vert \widehat{C} _{i+k_{n}}^{n}\right\vert ^{p}\right] \right] \leq K\frac{1}{k_{n}^{p}} \left( a_{n}\Delta _{n}^{\left( 2p-r\right) {\Greekmath 0124} +1-p}+K\right) , \end{equation} where the first inequality uses Taylor expansion and bounded derivatives of $ H$ and $G$, and the last transition uses ((ref)). The first term in ((ref)) satisfies, for $p\geq 1$, by Holder inequality, ((ref)) and ((ref)), \begin{eqnarray*} &&\sum_{i\in \overline{L}\left( n,T\right) :B_{i}}\mathrm{E}\left\vert {\Greekmath 0123} _{i}1_{\left\{ A_{i}\cap A_{i+k_{n}}\right\} }\right\vert \\ &\leq &\sum_{i\in \overline{L}\left( n,T\right) :B_{i}}\left( \mathrm{E} \left[ \left\vert {\Greekmath 0123} _{i}\right\vert ^{p}\right] \right) ^{1/p}\left( \mathrm{P}\left( A_{i}\cap A_{i+k_{n}}\right) \right) ^{\left( p-1\right) /p} \\ &\leq &\sum_{i\in \overline{L}\left( n,T\right) :B_{i}}K\left[ \Delta _{n}\times \left( \Delta _{n}^{\left( 2p-r\right) {\Greekmath 0124} +1-p}+1\right) \right] ^{1/p}\left[ \Delta _{n}^{1/2}\right] ^{\left( p-1\right) /p} \\ &=&\sum_{i\in \overline{L}\left( n,T\right) :B_{i}}K\Delta _{n}^{l\left( r,{\Greekmath 0124} \right) }. \end{eqnarray*} Since the number of terms in $\overline{L}\left( n,T\right) $ is bounded by $ Kk_{n}$ ($k_{n}$ arises due to overlapping blocks defining ${\Greekmath 0123} _{i}$ ), the first term in ((ref)) is $o_{p}\left( \Delta _{n}^{1/4}\right) $ if $l\left( r,{\Greekmath 0124} \right) >\frac{3}{4}$. To study $ l\left( r,{\Greekmath 0124} \right) $, we distinguish two cases, depending on whether $ \left( 2p-r\right) {\Greekmath 0124} +1-p\geq 0$ holds. Case 1. When $\left( 2p-r\right) {\Greekmath 0124} +1-p\geq 0$, $l\left( r,{\Greekmath 0124} \right) =\frac{1}{2p}\left( p+1\right) $, so $l\left( r,{\Greekmath 0124} \right) > \frac{3}{4}$ if $p<2.$ Case 2. When $\left( 2p-r\right) {\Greekmath 0124} +1-p<0$, $l\left( r,{\Greekmath 0124} \right) =\frac{1}{p}\left( \left( 2p-r\right) {\Greekmath 0124} +1-p\right) +\frac{p-1}{ 2p}$. We have $l\left( r,{\Greekmath 0124} \right) >\frac{3}{4}$ if ${\Greekmath 0124} >\frac{5p-6 }{4\left( 2p-r\right) }$. This is satisfied if we choose, for example, $ p=1.5 $. The last step in the proof of Theorem (ref) is to show that ((ref)) is true. In order to do that, we first prove that if there is a volatility jump on $\left( i\Delta _{n},\left( i+k_{n}\right) \Delta _{n}\right] $, then \begin{equation} P\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}\right \Vert <u_{n}^{\prime }\right) =o_{p}\left( \Delta _{n}^{1/4}\right) . \end{equation} Denote by $S$ the time of the volatility jump on $\left( i\Delta _{n},\left( i+k_{n}\right) \Delta _{n}\right] $, so the jump is $\Delta C_{S}$. Denote $ {\Greekmath 0118} _{n}\equiv \widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}-\Delta C_{S}$, so $\widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}=\Delta C_{S}+{\Greekmath 0118} _{n}$. We know ${\Greekmath 0118} _{n}=o_{p}\left( 1\right) $. We know that there exists ${\Greekmath 010F} $, independent of $i$ or $S$, such that $\left\Vert \Delta C\right\Vert > {\Greekmath 010F} $. We will first show that if there is a volatility jump on $\left( i\Delta _{n},\left( i+k_{n}\right) \Delta _{n}\right] $, for $s\geq 0$, it follows that \begin{equation} P\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}\right \Vert <u_{n}^{\prime }\right) \leq \frac{E\left( \left\Vert \widehat{C} _{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}-\Delta C_{S}\right\Vert ^{s}\right) }{\left( {\Greekmath 010F} /2\right) ^{s}}. \end{equation} To prove ((ref)), note that the reverse triangle inequality gives $\left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C} _{i-k_{n}}^{n}\right\Vert =\left\Vert \Delta C+{\Greekmath 0118} _{n}\right\Vert \geq \left\vert \left\Vert \Delta C\right\Vert -\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\vert $. Thus, \begin{eqnarray*} &&P\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C} _{i-k_{n}}^{n}\right\Vert <u_{n}^{\prime }\right) \\ &\leq &P\left( \left\vert \left\Vert \Delta C\right\Vert -\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\vert <u_{n}^{\prime }\right) \\ &\leq &P\left( \left\Vert {\Greekmath 0118} _{n}\right\Vert >\frac{{\Greekmath 010F} }{2}\right) , \end{eqnarray*} where the second inequality follows by distinguishing two cases, depending on whether $\left\Vert \Delta C\right\Vert \geq \left\Vert {\Greekmath 0118} _{n}\right\Vert $. Case 1:\ if $\left\Vert \Delta C\right\Vert \geq \left\Vert {\Greekmath 0118} _{n}\right\Vert $, $\left\{ \left\vert \left\Vert \Delta C\right\Vert -\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\vert <u_{n}^{\prime }\right\} =\left\{ \left\Vert \Delta C\right\Vert -\left\Vert {\Greekmath 0118} _{n}\right\Vert <u_{n}^{\prime }\right\} =\left\{ \left\Vert \Delta C\right\Vert -u_{n}^{\prime }<\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\} $, so we deduce $\left\{ {\Greekmath 010F} -u_{n}^{\prime }<\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\} $. For $n\,\ $large enough, this implies $\left\{ \left\Vert {\Greekmath 0118} _{n}\right\Vert >\frac{{\Greekmath 010F} }{2}\right\} $ since $u_{n}^{\prime }\rightarrow 0$. Case 2:\ if $\left\Vert \Delta C\right\Vert <\left\Vert {\Greekmath 0118} _{n}\right\Vert $, we have $P\left( \left\{ \left\vert \left\Vert \Delta C\right\Vert -\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\vert <u_{n}^{\prime }\right\} \cap \left\{ \left\Vert \Delta C\right\Vert <\left\Vert {\Greekmath 0118} _{n}\right\Vert \right\} \right) \leq P\left( \left\Vert {\Greekmath 0118} _{n}\right\Vert >\left\Vert \Delta C\right\Vert \right) \leq P\left( \left\Vert {\Greekmath 0118} _{n}\right\Vert >{\Greekmath 010F} \right) \leq P\left( \left\Vert {\Greekmath 0118} _{n}\right\Vert >\frac{{\Greekmath 010F} }{2}\right) $. Finally, ((ref)) follows by Markov's inequality. By ((ref)), we obtain, for $s\geq 2$, \begin{eqnarray} P\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}\right \Vert <u_{n}^{\prime }\right) &\leq &\frac{E\left( \left\Vert \widehat{C} _{i+k_{n}}^{n}-\widehat{C}_{i-k_{n}}^{n}-\Delta C\right\Vert ^{s}\right) }{ \left( {\Greekmath 010F} /2\right) ^{s}} \notag \\ &\leq &KE\left( \left\Vert \widehat{C}_{i-k_{n}}^{n}-C_{S-}\right\Vert ^{s}\right) +KE\left( \left\Vert \widehat{C}_{i+k_{n}}^{n}-C_{S}\right\Vert ^{s}\right) . \end{eqnarray} The first term in ((ref)) satisfies, for $s\geq 2,$ by ((ref)) and ((ref)) \begin{eqnarray*} E\left( \left\Vert \widehat{C}_{i-k_{n}}^{n}-C_{S-}\right\Vert ^{s}\right) &\leq &KE\left( \left\Vert \widehat{C}_{i-k_{n}}^{n}-\widehat{C} _{i-k_{n}}^{n\prime }\right\Vert ^{s}\right) +KE\left( \left\Vert \widehat{C} _{i-k_{n}}^{n\prime }-C_{S-}\right\Vert ^{s}\right) \\ &\leq &K_{q}a_{n}\Delta _{n}^{\left( 2s-r\right) {\Greekmath 0124} +1-s}+K\Delta _{n}^{s/4}. \end{eqnarray*} The second term in ((ref)) has the same bound by the same arguments as the first term. Choosing $s=2$ in the above, and taking into account that $\left( 2-r\right) {\Greekmath 0124} \geq \frac{3}{4}$ and $ {\Greekmath 0124} \geq \frac{3}{8}$, we obtain ((ref)). Given ((ref)), it is simple to obtain ((ref)) as follows. By ((ref)), if there is a jump on $\left( i\Delta _{n},\left( i+k_{n}\right) \Delta _{n} \right] $, we know $P\left( A_{i}\right) =o_{p}\left( \Delta _{n}^{1/4}\right) $, thus $\left( A_{i}\cap A_{i+k_{n}}\right) \leq P\left( A_{i}\right) =o_{p}\left( \Delta _{n}^{1/4}\right) $. Applying ((ref)) with $i+k_{n}$ instead of $i$, if there is a jump on $\left( \left( i+k_{n}\right) \Delta _{n},\left( i+2k_{n}\right) \Delta _{n}\right] $, $P\left( A_{i+k_{n}}\right) =o_{p}\left( \Delta _{n}^{1/4}\right) $. Thus, $P\left( A_{i}\cap A_{i+k_{n}}\right) \leq P\left( A_{i+k_{n}}\right) =o_{p}\left( \Delta _{n}^{1/4}\right) $. We conclude that if there is a jump on $\left( i\Delta _{n},\left( i+2k_{n}\right) \Delta _{n}\right] $, i.e., event $B_{i}$ is true, then $\left( A_{i}\cap A_{i+k_{n}}\right) \leq P\left( A_{i+k_{n}}\right) =o_{p}\left( \Delta _{n}^{1/4}\right) $. This concludes the proof of ((ref)) and hence Theorem (ref). \subsection{Proof of Theorem (ref)} To show this result, let us define the functions \begin{align*} R(x,y)& =\sum_{g,h,a,b=1}^{d}\Big(\partial _{gh}H\partial _{ab}G\big)(x)\big( y^{gh}-x^{gh}\Big)\Big(y^{ab}-x^{ab}\Big) \\ S(x,y)& =\Big(H(y)-H(x)\Big)\Big(G(y)-G(x)\Big) \\ U(x)& =\sum_{g,h,a,b=1}^{d}\Big(\partial _{gh}H\partial _{ab}G\Big)(x)\Big( x^{ga}x^{hb}+x^{gb}x^{ha}\Big), \end{align*} for any $\mathbb{R}^{d}\times \mathbb{R}^{d}$ matrices $x$ and $y$. The following decompositions hold, \begin{align*} & \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{AN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime AN} \\ & =\frac{3}{2k_{n}}\sum_{i\in L\left( n,T\right) }\Big[\big(S( \widehat{C}_{i}^{n},\widehat{C}_{i+k_{n}}^{n})-S(\widehat{C}_{i}^{^{\prime }n},\widehat{C}_{i+k_{n}}^{^{\prime }n})\big)-\frac{2}{k_{n}}\big(U(\widehat{ C}_{i}^{n})-U(\widehat{C}_{i}^{^{\prime }n})\big)\Big], \\ & \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{LIN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime LIN} \\ & =\frac{3}{2k_{n}}\sum_{i\in L\left( n,T\right) }\Big[\big(R( \widehat{C}_{i}^{n},\widehat{C}_{i+k_{n}}^{n})-R(\widehat{C}_{i}^{^{\prime }n},\widehat{C}_{i+k_{n}}^{^{\prime }n})\big)-\frac{2}{k_{n}}\big(U(\widehat{ C}_{i}^{n})-U(\widehat{C}_{i}^{^{\prime }n})\big)\Big]. \end{align*} Since $H$ and $G$ are three times continuously differentiable with bounded derivatives, the functions $R$ and $S$ are continuously differentiable and satisfy \begin{eqnarray} \Vert \partial J(x,y)\Vert &\leq &K\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for}J\in \{S,R\}, \\ \Vert \partial U(x)\Vert &\leq &K, \end{eqnarray} where $\partial J$ (respectively, $\partial U$) is a vector that collects the first order partial derivatives of the function $J$ (respectively, $U$) with respect to all the elements of $(x,y)$ (respectively, $x$). Using the Taylor expansion, ((ref)) and ((ref)), it holds that, for $J\in \{S,R\}$, \begin{align*} |J(\widehat{C}_{i}^{n},\widehat{C}_{i+k_{n}}^{n})-J(\widehat{C} _{i}^{^{\prime }n},\widehat{C}_{i+k_{n}}^{^{\prime }n})|& \leq K(\Vert \widehat{C}_{i}^{n}-\widehat{C}_{i}^{^{\prime }n}\Vert +\Vert \widehat{C} _{i+k_{n}}^{n}-\widehat{C}_{i+k_{n}}^{^{\prime }n}\Vert ) \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \\ |U(\widehat{C}_{i}^{n})-U(\widehat{C}_{i}^{^{\prime }n})|& \leq K(\Vert \widehat{C}_{i}^{n}-\widehat{C}_{i}^{^{\prime }n}\Vert ). \end{align*} By equation ((ref)), the following condition is sufficient for Theorem (ref) to hold: \begin{equation*} (2-r){\Greekmath 0124} -\frac{3}{4}\geq 0. \end{equation*} The above condition follows from our assumptions of Theorem (ref). Using the fact that $0<{\Greekmath 0124} <\frac{1}{2}$, we can see that Theorem (ref) holds when $3/4(2-r)\leq {\Greekmath 0124} <\frac{1}{2}$, which completes the proof. \subsection{Proof of Theorem (ref)} Note that we have \begin{align*} & \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime LIN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }=\frac{3}{2k_{n}} \sum_{g,h,a,b=1}^{d}\sum_{i\in L\left( n,T\right) }{\Greekmath 0120} _{i}^{n}(g,h,a,b), \\ & \sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\prime AN}-\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }=\frac{3}{2k_{n}} \sum_{i\in L\left( n,T\right) }\Big({\Greekmath 011F} _{i}^{n}-\sum_{g,h,a,b=1}^{d}\big( \partial _{gh}H\partial _{ab}G\big)(C_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}\Big), \end{align*} with \begin{align*} & {\Greekmath 0120} _{i}^{n}(g,h,a,b)=\Big(\big(\partial _{gh}H\partial _{ab}G\big)( \widehat{C}_{i}^{^{\prime }n})-\big(\partial _{gh}H\partial _{ab}G\big) (C_{i}^{n})\Big){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}, \\ & {\Greekmath 011F} _{i}^{n}=\Big(H(\widehat{C}_{i+k_{n}}^{^{\prime }n})-H(\widehat{C} _{i}^{^{\prime }n})\Big)\Big(G(\widehat{C}_{i+k_{n}}^{^{\prime }n})-G( \widehat{C}_{i}^{^{\prime }n})\Big). \end{align*} By Taylor expansion, we have \begin{align*} & \big(\partial _{gh}S\partial _{ab}G\big)(\widehat{C}_{i}^{^{\prime }n})- \big(\partial _{gh}S\partial _{ab}G\big)(C_{i}^{n})=\sum_{x,y=1}^{d}\Big( \partial _{xy,gh}^{2}S\partial _{ab}G+\partial _{xy,ab}^{2}G\partial _{gh}S \Big)(C_{i}^{n}){\Greekmath 0117} _{i}^{n,xy} \\ & +\frac{1}{2}\sum_{j,k,x,y=1}^{d}\Big(\partial _{jk,xy,gh}^{3}S\partial _{ab}G+\partial _{xy,gh}^{2}S\partial _{jk,ab}^{2}G+\partial _{jk,xy,ab}^{3}G\partial _{gh}S+\partial _{xy,ab}^{2}G\partial _{jk,gh}^{2}S \Big)(\widetilde{c}_{i}^{n}){\Greekmath 0117} _{i}^{n,xy}{\Greekmath 0117} _{i}^{n,jk} \end{align*} and \begin{eqnarray*} &&S(\widehat{C}_{i+k_{n}}^{^{\prime }n})-S(\widehat{C}_{i}^{^{\prime }n})=\sum_{gh}\partial _{gh}S(C_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}+\sum_{j,k,g,h}\partial _{jk,gh}^{2}S(C_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0117} _{i}^{n,jk} \\ &&+\frac{1}{2}\sum_{x,y,g,h}\partial _{xy,gh}^{2}S(C_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,xy}+\frac{1}{2}\sum_{x,y,j,k,g,h}\partial _{xy,jk,gh}^{3}S(CC_{i}^{n,S}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0117} _{i}^{n,xy}{\Greekmath 0117} _{i}^{n,jk} \\ &&+\frac{1}{6}\sum_{j,k,x,y,g,h}\partial _{jk,xy,gh}^{3}S(C_{i}^{n,S}){\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,xy}, \end{eqnarray*} for $S\in \{H,G\}$, $\widetilde{c}_{i}^{n}={\Greekmath 0119} C_{i}^{n}+(1-{\Greekmath 0119} )\widehat{C} _{i}^{^{\prime }n}$, $C_{i}^{n,S}={\Greekmath 0119} _{S}\widehat{C}_{i}^{^{\prime }n}+(1-{\Greekmath 0119} _{S})\widehat{C}_{i+k_{n}}^{^{\prime }n}$, $CC_{i}^{n,S}={\Greekmath 0116} _{S}C_{i}^{n}+(1-{\Greekmath 0116} _{S})\widehat{C}_{i}^{^{\prime }n}$ for ${\Greekmath 0119} ,{\Greekmath 0119} _{H},{\Greekmath 0116} _{H},{\Greekmath 0119} _{G},{\Greekmath 0116} _{G}\in \lbrack 0,1]$. Although $\widetilde{c} _{i}^{n}$ and ${\Greekmath 0119} $ depend on $g,h,a,$ and $b$, we do not emphasize this in our notation to simplify the exposition.\newline By (4.10) in jacodrosenbaum13 we have \begin{equation} \mathbb{E}\Big(\Big\|{\Greekmath 010B} _{i}^{n}\Big\|^{q}\Big|\mathcal{F}_{(i-1)\Delta_n}\Big) \leq K_{q}\Delta _{n}^{q}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all}q\geq 0 \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\mathbb{E}\Big(\Big|\sum_{j=0}^{k_{n}-1} {\Greekmath 010B} _{i+j}^{n}\Big|^{q}\big|\mathcal{F}_{(i-1)\Delta_n}\Big)\leq K_{q}\Delta _{n}^{q}k_{n}^{q/2}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for}q\geq 2. \end{equation} Combining ((ref)), ((ref)), ((ref)) with $Z=C$ and the H\"{o}lder inequality yields for $q\geq 2$, for $i\in L\left( n,T\right) $ \begin{equation} \mathbb{E}\Big(\Big\|{\Greekmath 0117} _{i}^{n}\Big\|^{q}\Big|\mathcal{F}_{(i-1)\Delta_n}\Big)\leq K_{q}\Delta ^{q/4},\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\mathbb{E}\Big(\Big\| {\Greekmath 0115} _{i}^{n}\Big\|^{q}\Big|\mathcal{F}_{(i-1)\Delta_n}\Big)\leq K_{q}\Delta ^{q/4}. \end{equation} The bound in the first equation of ((ref)) is tighter than that in (4.11) of jacodrosenbaum-sqrtn due to the absence of volatility jumps. This tighter bound will be useful later in deriving the asymptotic distribution for the approximated estimator. By the boundedness of $C_{t}$ and the derivatives of $H$ and $G$, \begin{equation} \Big|\big(\partial _{jk,xy,ab}^{3}G\partial _{gh}H+\partial _{xy,gh}^{2}H\partial _{jk,ab}^{2}G\big)(\widetilde{c}_{i}^{n}){\Greekmath 0117} _{i}^{n,xy}{\Greekmath 0117} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}\Big|\leq K\Vert {\Greekmath 0117} _{i}^{n}\Vert ^{2}\Vert {\Greekmath 0115} _{i}^{n}\Vert ^{2}. \end{equation} Using the Taylor expansion, we have \begin{align*} & {\Greekmath 011F} _{i}^{n}-\sum_{g,h,a,b}(\partial _{gh}H\partial _{ab}G)(C_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}= \\ & \sum_{g,h,a,b,j,k}(\partial _{gh}H\partial _{jk,xy}^{2}G+\partial _{gh}G\partial _{jk,xy}^{2}H)(C_{i}^{n})({\Greekmath 0115} _{i}^{n,gh}+\frac{1}{2}{\Greekmath 0117} _{i}^{n,gh}){\Greekmath 0115} _{i}^{n,ab}{\Greekmath 0115} _{i}^{n,jk}+{\Greekmath 0127} _{i}^{n},\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ and} \\ & \sum_{g,h,a,b}\big(\partial _{gh}H\partial _{ab}G\big)(\widehat{C} _{i}^{^{\prime }n})-\big(\partial _{gh}H\partial _{ab}G\big)(C_{i}^{n})= \\ & \sum_{g,h,a,b,x,y}(\partial _{gh}H\partial _{ab,xy}^{2}G+\partial _{ab}G\partial _{gh,xy}^{2}G)(C_{i}^{n})({\Greekmath 0117} _{i}^{n,xy}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}+{\Greekmath 010E} _{i}^{n} \end{align*} with $\mathbb{E}(|{\Greekmath 0127} _{i}^{n}|\big|\mathcal{F}_{i}^{n})\leq K\Delta _{n} $ and $\mathbb{E}(|{\Greekmath 010E} _{i}^{n}|\big|\mathcal{F}_{i}^{n})\leq K\Delta _{n} $ which follow by the Cauchy-Schwartz inequality together with equation ((ref)). Given that $k_{n}={\Greekmath 0112} (\Delta _{n})^{-1/2}$, the previous inequalities imply \begin{equation*} \frac{3\Delta _{n}^{-1/4}}{2k_{n}}\sum_{i\in L\left( n,T\right) }{\Greekmath 0127} _{i}^{n}\overset{\mathbb{P}}{\Longrightarrow }0\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\frac{3\Delta _{n}^{-1/4}}{2k_{n}}\sum_{i\in L\left( n,T\right) }{\Greekmath 010E} _{i}^{n}\overset{\mathbb{P}}{\Longrightarrow }0. \end{equation*} Therefore, it suffices to show that \begin{equation} \frac{3\Delta _{n}^{-1/4}}{2k_{n}}\sum_{i\in L\left( n,T\right) }\sum_{g,h,a,b,j,k}(\partial _{gh}H\partial _{jk,ab}^{2}G+\partial _{gh}H\partial _{jk,ab}^{2}G)(C_{i}^{n}){\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}{\Greekmath 0115} _{i}^{n,jk}\overset{\mathbb{P}}{\longrightarrow }0, \end{equation} \begin{equation} \frac{3\Delta _{n}^{-1/4}}{2k_{n}}\sum_{i\in L\left( n,T\right) }\sum_{g,h,a,b,j,k}(\partial _{gh}H\partial _{jk,ab}^{2}G+\partial _{gh}H\partial _{jk,ab}^{2}G)(C_{i}^{n}){\Greekmath 0117} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}{\Greekmath 0115} _{i}^{n,jk}\overset{\mathbb{P}}{\longrightarrow }0. \end{equation} These results hold by the bounds in Lemma (ref). \subsection{Proof of Theorem (ref)} In Section (ref), to simplify the notational burden, we adopt the following strategy. Instead of studying $\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A\right) }$, we work with all indices $i$ , i.e., $\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{\left( A\right) }$, together with the assumption that there are no volatility jumps. The difference between the two quantities is $o_{p}\left( \Delta _{n}^{1/4}\right) $ because in the absence of volatility jumps, ${\Greekmath 0123} _{i}^{\left( A\right) }$ satisfies the bound in equation ((ref)). Recall the decomposition from from (ref), \begin{equation} {\Greekmath 0123} _{i}^{\left( A\right) } = {\Greekmath 0123} _{i}^{\left( A1\right)} - {\Greekmath 0123} _{i}^{\left( A2\right) }. \end{equation} Given the boundedness of the derivatives of $H$ and $G$ and the fact that $ k_{n}={\Greekmath 0112} (\Delta _{n})^{-1/2}$, by Theorem 2.2 in jacodrosenbaum-sqrtn we have \begin{equation*} \frac{1}{\sqrt{\Delta _{n}}}\Bigg(\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{\left( A2\right) }-\frac{3}{{\Greekmath 0112} ^{2}} \sum_{g,h,a,b=1}^{d}\int_{0}^{T}\big(\partial _{gh}H\partial _{ab}G\big) (C_{t})(C_{t}^{ga}C_{t}^{hb}+C_{t}^{gb}C_{t}^{ha})dt\Bigg)=O_{p}(1), \end{equation*} which yields \begin{equation*} \frac{1}{\Delta _{n}^{1/4}}\Bigg(\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{\left( A2\right) }-\frac{3}{{\Greekmath 0112} ^{2}} \sum_{g,h,a,b=1}^{d}\int_{0}^{T}\big(\partial _{gh}H\partial _{ab}G\big) (C_{t})(C_{t}^{ga}C_{t}^{hb}+C_{t}^{gb}C_{t}^{ha})dt\Bigg)\overset{\mathbb{P} }{\longrightarrow }0. \end{equation*} Using the multivariate quantities defined in Section (ref), we can show that the following decompositions hold: \begin{align*} \widehat{C}_{i}^{^{\prime }n}& =C_{i-1}^{n}+\frac{1}{k_{n}} \sum_{j=0}^{k_{n}-1}\sum_{u=1}^{2}\overline{{\Greekmath 0122} }(u)_{j}^{n}{\Greekmath 0110} (u)_{i+j}^{n},\widehat{C}_{i+k_{n}}^{^{\prime }n}-\widehat{C} _{i}^{^{\prime }n}=\frac{1}{k_{n}}\sum_{j=0}^{2k_{n}-1}\sum_{u=1}^{2} {\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0110} (u)_{i+j}^{n}, \\ {\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}& =\frac{1}{k_{n}^{2}} \sum_{u=1}^{2}\sum_{v=1}^{2}\Bigg(\sum_{j=0}^{2k_{n}-1}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+j}^{n,ab} \\ & +\sum_{j=0}^{2k_{n}-2}\sum_{q=j+1}^{2k_{n}-1}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{q}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+q}^{n,ab}+\sum_{j=1}^{2k_{n}-1}\sum_{q=0}^{j-1}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{q}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+q}^{n,ab}\Bigg). \end{align*} Changing the order of the summation in the last term yields \begin{align*} {\Greekmath 0115} _{i}^{n,gh}{\Greekmath 0115} _{i}^{n,ab}& =\frac{1}{k_{n}^{2}} \sum_{u=1}^{2}\sum_{v=1}^{2}\Bigg(\sum_{j=0}^{2k_{n}-1}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+j}^{n,ab} \\ & +\sum_{j=0}^{2k_{n}-2}\sum_{q=j+1}^{2k_{n}-1}{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{q}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+q}^{n,ab}+\sum_{j=0}^{2k_{n}-2}\sum_{q=j+1}^{2k_{n}-1}{\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0122} (u)_{q}^{n}{\Greekmath 0110} (v)_{i+j}^{n,ab}{\Greekmath 0110} (u)_{i+q}^{n,gh}\Bigg). \end{align*} Therefore, we can further rewrite $\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{\left( A1\right) }$ as \begin{align*} & \sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{\left( A1\right) }=\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A11\right) }+\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A12\right) }+\sum_{i\in L\left( n,T\right) }{\Greekmath 0123} _{i}^{\left( A13\right) },\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with} \\ & \sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}{\Greekmath 0123} _{i}^{\left( A1w\right) }=\sum_{g,h,a,b=1}^{d}\sum_{u,v=1}^{2}\widehat{A1w} (H,gh,u;G,ab,v)_{T}^{n},w=1,2,3, \end{align*} where \begin{align*} & \widehat{A11}(H,gh,u;G,ab,v)_{T}^{n}=\frac{3}{2k_{n}^{3}} \sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}\sum_{j=0}^{2k_{n}-1}(\partial _{gh}H\partial _{ab}G)(C_{i-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+j}^{n,ab}, \\ & \widehat{A12}(H,gh,u;G,ab,v)_{T}^{n}=\frac{3}{2k_{n}^{3}} \sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}\sum_{j=0}^{2k_{n}-2}\sum_{q=j+1}^{2k_{n}-1}(\partial _{gh}H\partial _{ab}G)(C_{i-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{q}^{n}{\Greekmath 0110} (u)_{i+j}^{n,gh}{\Greekmath 0110} (v)_{i+q}^{n,ab}, \\ & \widehat{A13}(H,gh,u;G,ab,v)_{T}^{n}=\frac{3}{2k_{n}^{3}} \sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}\sum_{j=0}^{2k_{n}-2}\sum_{q=j+1}^{2k_{n}-1}(\partial _{gh}H\partial _{ab}G)(C_{i-1}^{n}){\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0122} (u)_{q}^{n}{\Greekmath 0110} (v)_{i+j}^{n,ab}{\Greekmath 0110} (u)_{i+q}^{n,gh}, \end{align*} where we clearly have $\widehat{A13}(H,gh,u;G,ab,v)_{T}^{n}=\widehat{A12} (G,ab,v;H,gh,u)_{T}^{n}.$ By a change of the order of the summation, \begin{align*} \widehat{A11}(H,gh,u;G,ab,v)_{T}^{n}& =\frac{3}{2k_{n}^{3}} \sum_{i=1}^{[T/\Delta _{n}]}\sum_{j=0\vee (i+2k_{n}-1-[T/\Delta _{n}])}^{(2k_{n}-1)\wedge (i-1)}(\partial _{gh}H\partial _{ab}G) \\ & \times (C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}, \\ \widehat{A12}(H,gh,u;G,ab,v)_{T}^{n}& =\frac{3}{2k_{n}^{3}} \sum_{i=2}^{[T/\Delta _{n}]}\sum_{m=1}^{(i-1)\wedge (2k_{n}-1)}\sum_{j=0\vee (i+2k_{n}-1-m-[T/\Delta _{n}])}^{(2k_{n}-m-1)\wedge (i-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-1-j-m}^{n}) \\ & \times {\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n}{\Greekmath 0110} _{gh}(u)_{i-m}^{n}{\Greekmath 0110} _{ab}(v)_{i}^{n}. \end{align*} Now, set \begin{align*} \widetilde{A11}(H,gh,u;G,ab,v)_{T}^{n}& =\frac{3}{2k_{n}^{3}} \sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\sum_{j=0}^{2k_{n}-1}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}{\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}, \\ \widetilde{A12}(H,gh,u;G,ab,v)_{T}^{n}& =\frac{3}{2k_{n}^{3}} \sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\sum_{m=1}^{(i-1)\wedge (2k_{n}-1)}\sum_{j=0}^{(2k_{n}-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1-m}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n} \\ & \times {\Greekmath 0110} _{gh}(u)_{i-m}^{n}{\Greekmath 0110} _{ab}(v)_{i}^{n}. \end{align*} We show below that the following results hold: \begin{equation} \frac{1}{\Delta _{n}^{1/4}}\Big(\widehat{A1w}(H,gh,u;G,ab,v)_{T}^{n}- \widetilde{A1w}(H,gh,u;G,ab,v)_{T}^{n}\Big)\overset{\mathbb{P}}{ \longrightarrow }0 \end{equation} \begin{equation} \frac{1}{\Delta _{n}^{1/4}}\Big(\widetilde{A1w}(H,gh,u;G,ab,v)_{T}^{n}- \overline{A1w}(H,gh,u;G,ab,v)_{T}^{n}\Big)\overset{\mathbb{P}}{ \longrightarrow }0 \end{equation} \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all} $(H,gh,u,G,ab,v)~\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}~w=1,2.$ \subsubsection{Proof of Equation ((ref)) for $w=1$} To prove this result, first, notice that the ${\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}$ are scaled by random variables rather that constant real numbers. Next, observe that we can write \begin{align*} & \widehat{A11}-\widetilde{A11}=\widetilde{\widehat{A11}}(1)+\widetilde{ \widehat{A11}}(2)+\widetilde{\widehat{A11}}(3)\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{with} \\ & \widetilde{\widehat{A11}}(1)=\sum_{i=1}^{(2k_{n}-1)\wedge \lbrack T/\Delta _{n}]}\Bigg(\frac{3}{2k_{n}^{3}}\sum_{j=0\vee (i+2k_{n}-1-[T/\Delta _{n}])}^{(2k_{n}-1)\wedge (i-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}\Bigg) {\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}, \\ & \widetilde{\widehat{A11}}(2)=\sum_{i=[T/\Delta _{n}]-2k_{n}+2}^{[T/\Delta _{n}]}\frac{3}{2k_{n}^{3}}\Bigg(\sum_{j=0\vee (i+2k_{n}-1-[T/\Delta _{n}])}^{(2k_{n}-1)\wedge (i-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n} \\ & -\sum_{j=0}^{(2k_{n}-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}\Bigg) {\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}, \\ & \widetilde{\widehat{A11}}(3)=\sum_{i=2k_{n}}^{[T/\Delta _{n}]-2k_{n}+1} \frac{3}{2k_{n}^{3}}\Bigg(\sum_{j=0\vee (i+2k_{n}-1-[T/\Delta _{n}])}^{(2k_{n}-1)\wedge (i-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n} \\ & -\sum_{j=0}^{(2k_{n}-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}\Bigg) {\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}. \end{align*} It is easy to see that $\widetilde{\widehat{A12}}(3)=0$. Using equation ((ref)) with $Z=c$ and equation ((ref)), we obtain \begin{equation} \mathbb{E}(\Vert {\Greekmath 0110} (1)_{i}^{n}\Vert ^{q}|\mathcal{F}_{i-1}^{n})\leq K_{q},\mathbb{E}(\Vert {\Greekmath 0110} (2)_{i}^{n}\Vert ^{q}|\mathcal{F} _{i-1}^{n})\leq K_{q}\Delta _{n}^{q/2}. \end{equation} By the boundedness of the derivatives of $H$ and $G$, the random quantities $\Big(\frac{3}{2k_{n}^{3}}\sum_{j=0\vee (i+2k_{n}-1-[T/\Delta _{n}])}^{(2k_{n}-1)\wedge (i-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}\Big)$ and\newline $\frac{3}{2k_{n}^{3}}\sum_{j=0}^{(2k_{n}-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}$ are $ \mathcal{F}_{i-1}^{n}-$ measurable and are bounded by $\widetilde{{\Greekmath 0115} } _{u,v}^{n}$ defined as \begin{equation*} \widetilde{{\Greekmath 0115} }_{u,v}^{n}= \begin{cases} K & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \\ K/k_{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,2),(2,1) \\ K/k_{n}^{2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,1). \end{cases} \end{equation*} Similarly, the quantity \begin{equation*} \frac{3}{2k_{n}^{3}}\Bigg(\sum_{j=0\vee (i+2k_{n}-1-[T/\Delta _{n}])}^{(2k_{n}-1)\wedge (i-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}-\sum_{j=0}^{(2k_{n}-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}\Bigg), \end{equation*} is $\mathcal{F}_{i-1}^{n}-$ measurable and bounded by $2\widetilde{{\Greekmath 0115} } _{u,v}^{n}$. Note also that, by equation ((ref)) and the Cauchy Schwartz inequality, we have \begin{align*} \mathbb{E}(|{\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|\big|\mathcal{F} _{i-1}^{n})& \leq \mathbb{E}(\Vert {\Greekmath 0110} (u)_{i}^{n}\Vert ^{2}|\mathcal{F} _{i-1}^{n})^{1/2}\mathbb{E}(\Vert {\Greekmath 0110} (v)_{i}^{n}\Vert ^{2}|\mathcal{F} _{i-1}^{n})^{1/2} \\ & \leq \begin{cases} K\Delta _{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \\ K\Delta _{n}^{1/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,2),(2,1) \\ K & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,1). \end{cases} \end{align*} The above bounds, together with the fact that $k_{n}={\Greekmath 0112} \Delta _{n}^{-1/2}$, imply $\mathbb{E}(|\widetilde{\widehat{A11}}(1)|)\leq K\Delta _{n}^{1/2}$ and $\mathbb{E}(|\widetilde{\widehat{A11}}(2)|)\leq K\Delta _{n}^{1/2}$ for all $(u,v)$. These two results together imply $\widetilde{ \widehat{A11}}(1)=o(\Delta _{n}^{-1/4})$ and $\widetilde{\widehat{A11}} (2)=o(\Delta _{n}^{-1/4})$, which yields the result. \subsubsection{Proof of Equation ((ref)) for $w=2$} First, observe that $\widehat{A12}-\widetilde{A12}=\widetilde{\widehat{A12}} (1)+\widetilde{\widehat{A12}}(2)$, with \begin{align*} & \widetilde{\widehat{A12}}(1)=\sum_{i=2}^{(2k_{n}-1)\wedge \lbrack T/\Delta _{n}]}\Bigg(\sum_{m=1}^{(i-1)}\frac{3}{2k_{n}^{3}}\Big(\sum_{j=0\vee (i+2k_{n}-1-m-[T/\Delta _{n}])}^{(2k_{n}-m-1)\wedge (i-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-1-j-m}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n}\Big) \\ & \times {\Greekmath 0110} _{gh}(u)_{i-m}^{n}\Bigg){\Greekmath 0110} _{ab}(v)_{i}^{n}, \\ & \widetilde{\widehat{A12}}(2)=\sum_{i=[T/\Delta _{n}]-2k_{n}+2}^{[T/\Delta _{n}]}\Bigg(\sum_{m=1}^{(i-1)\wedge (2k_{n}-1)}\Big(\frac{3}{2k_{n}^{3}} \sum_{j=0\vee (i+2k_{n}-1-m-[T/\Delta _{n}])}^{(2k_{n}-m-1)\wedge (i-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-1-j-m}^{n}){\Greekmath 0122} (u)_{j}^{n} \\ & \times {\Greekmath 0122} (v)_{j+m}^{n}\Big) -\sum_{j=0}^{(2k_{n}-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-1-j-m}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n}\Big) {\Greekmath 0110} _{gh}(u)_{i-m}^{n}\Bigg){\Greekmath 0110} _{ab}(v)_{i}^{n}. \end{align*} Notice that the quantity \begin{equation*} {\Greekmath 0114} _{i}^{m,n}=\frac{3}{2k_{n}^{3}}\Big(\sum_{j=0\vee (i+2k_{n}-1-m-[T/\Delta _{n}])}^{(2k_{n}-m-1)\wedge (i-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-1-j-m}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n}\Big) \end{equation*} is $\mathcal{F}_{i-m-1}^{n}$ measurable and bounded by $\widetilde{{\Greekmath 0115} } _{u,v}^{n}$. Let \begin{equation*} {\Greekmath 0114} _{i}^{n}=\sum_{m=1}^{(i-1)}\frac{3}{2k_{n}^{3}}\Big(\sum_{j=0\vee (i+2k_{n}-1-m-[T/\Delta _{n}])}^{(2k_{n}-m-1)\wedge (i-m-1)}(\partial _{gh}H\partial _{ab}G)(C_{i-1-j-m}^{n}){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n}\Big){\Greekmath 0110} _{gh}(u)_{i-m}^{n}. \end{equation*} It follows that ${\Greekmath 0114} _{i}^{n}$ is $\mathcal{F}_{i-1}^{n}$-measurable and we have \begin{eqnarray*} \mathbb{E}(|{\Greekmath 0114} _{i}^{m,n}|^{z}\big|\mathcal{F}_{0}) &\leq &(\widetilde{ {\Greekmath 0115} }_{u,v}^{n})^{z}, \\ |\mathbb{E}({\Greekmath 0110} (u)_{i-m}^{n}|\mathcal{F}_{i-m-1})| &\leq & \begin{cases} K\sqrt{\Delta _{n}} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=1 \\ K\Delta _{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=2 \end{cases} , \\ \mathbb{E}(\Vert {\Greekmath 0110} (u)_{i-m}^{n}\Vert ^{z}|\mathcal{F}_{i-m-1}) &\leq & \begin{cases} K_{z} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=1 \\ K_{z}\Delta _{n}^{z/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=2 \end{cases} . \end{eqnarray*} Using Lemma (ref), we deduce that for $z\geq 2$, \begin{equation*} \mathbb{E}(|{\Greekmath 0114} _{i}^{n}|^{z})\leq \begin{cases} K_{z}(\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{z}k_{n}^{z/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} u=1 \\ K_{z}(\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{z}/k_{n}^{z/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} u=2 \end{cases} \leq \begin{cases} K_{z}/k_{n}^{-3z/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}v=1 \\ K_{z}k_{n}^{-z/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}v=2 \end{cases} . \end{equation*} Using the above result, we obtain $\frac{1}{\Delta _{n}^{1/4}}\widetilde{ \widehat{A12}}(1)\overset{\mathbb{P}}{\Rightarrow }0$. A similar argument yields $\frac{1}{\Delta _{n}^{1/4}}\widetilde{\widehat{A12}}(2)\overset{ \mathbb{P}}{\Rightarrow }0$, which completes the proof of the equation ((ref)) for $w=2$. \subsubsection{Proof of Equation ((ref)) for $w=1$} Define \begin{equation*} \Theta (u,v)_{0}^{(C),i,n}=\frac{3}{2k_{n}^{3}} \sum_{j=0}^{2k_{n}-1}\Big((\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n})-(\partial _{gh}H\partial _{ab}G)(C_{i-2k_{n}}^{n}) \Big){\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j}^{n}. \end{equation*} By Taylor expansion, boundedness of the derivatives of $H$ and $G$, and using ((ref)) with $Z=c$, we have \begin{align*} & \Big|\mathbb{E}\Big((\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n})-(\partial _{gh}H\partial _{ab}G)(C_{i-2k_{n}}^{n}) \big|\mathcal{F}_{i-2k_{n}}^{n}\Big)\Big|\leq K(k_{n}\Delta _{n})\leq K\sqrt{ \Delta _{n}} \\ & \mathbb{E}(|(\partial _{gh}H\partial _{ab}G)(C_{i-j-1}^{n})-(\partial _{gh}H\partial _{ab}G)(C_{i-2k_{n}}^{n})|^{q}|\mathcal{F}_{i-2k_{n}}^{n})| \leq K(k_{n}\Delta _{n})^{q/2}\leq K\Delta _{n}^{q/4}, \end{align*} for $q\geq 2$ and for $j=0,\ldots ,2k_{n}-1$. Next, observe that $\Theta (u,v)_{0}^{(C),i,n}$ is $\mathcal{F}_{i-1}^{n}$ -measurable and satisfies $ |\Theta (u,v)_{0}^{(C),i,n}|\leq \widetilde{{\Greekmath 0115} }_{u,v}^{n}$, $|\mathbb{E }\Big(\Theta (u,v)_{0}^{(C),i,n}|\mathcal{F}_{i-2k_{n}}^{n}\Big)|\leq K\Delta _{n}^{1/2}\widetilde{{\Greekmath 0115} }_{u,v}^{n}$ and $\mathbb{E}\Big( |\Theta (u,v)_{0}^{(C),i,n}|^{q}\big|\mathcal{F}_{i-2k_{n}}^{n}\Big)\leq K_{q}\Delta _{n}^{q/4}(\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{q}$ where the latter follows from the H\"{o}lder inequality. We aim to prove that \begin{equation*} \widehat{E}=\frac{1}{\Delta _{n}^{1/4}}\Bigg[\sum_{i=2k_{n}}^{[T/\Delta _{n}]}\Theta (u,v)_{0}^{(C),i,n}{\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab} \Bigg] \end{equation*} converges to zero in probability for any $H$, $G$, $g$, $h$, $a$, and $b$ with $u,v=1,2$. \newline To show this result, we first introduce the following quantities: \begin{align*} & \widehat{E}(1)=\frac{1}{\Delta _{n}^{1/4}}\Bigg[\sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\Theta (u,v)_{0}^{(C),i,n}\mathbb{E}({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|\mathcal{F}_{i-1}^{n})\Bigg] \\ & \widehat{E}(2)=\frac{1}{\Delta _{n}^{1/4}}\Bigg[\sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\Theta (u,v)_{0}^{(C),i,n}\big({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}-\mathbb{E}({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|\mathcal{F }_{i-1}^{n})\big)\Bigg], \end{align*} with $\widehat{E}=\widehat{E}(1)+\widehat{E}(2)$. By Cauchy-Schwartz inequality, we have \begin{equation*} \mathbb{E}(|{\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|^{q})\leq (\widehat{ {\Greekmath 0115} }_{u,v}^{n})^{q/2},\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{where}\widehat{{\Greekmath 0115} } _{u,v}^{n}= \begin{cases} K & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,1) \\ K\Delta _{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,2),(2,1) \\ K\Delta _{n}^{2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \end{cases} \end{equation*} Since ${\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}$ is $\mathcal{F}_{i}^{n}$ -measurable,\newline the martingale property of ${\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}-\mathbb{ E}({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|\mathcal{F}_{i-1}^{n})$ implies, for all $(u,v)$, \begin{equation*} \mathbb{E}(|\widehat{E}(2)|^{2})\leq K\Delta _{n}^{-3/2}(\Delta _{n}^{1/4} \widetilde{{\Greekmath 0115} }_{u,v}^{n})^{2}\widehat{{\Greekmath 0115} }_{u,v}^{n}\leq K\Delta _{n}. \end{equation*} The latter inequality implies $\widehat{E}(2)\overset{\mathbb{P}}{ \Rightarrow }0$ for all $(u,v)$. It remains to show that $\widehat{E}(1) \overset{\mathbb{P}}{\Rightarrow }0$.\newline Here, we recall some bounds under Assumption (ref), \begin{align} & |\mathbb{E}({\Greekmath 0110} (1)_{i}^{n,gh}{\Greekmath 0110} (2)_{i}^{n,ab}|\mathcal{F} _{i-1}^{n})|\leq K\Delta _{n}, \\ & |\mathbb{E}({\Greekmath 0110} (1)_{i}^{n,gh}{\Greekmath 0110} (1)_{i}^{n,ab}|\mathcal{F} _{i-1}^{n})-\big(C_{i-1}^{n,ga}C_{i-1}^{n,hb}+C_{i-1}^{n,gb}C_{i-1}^{n,ha} \big)|\leq K\Delta _{n}^{1/2}, \\ & |\mathbb{E}({\Greekmath 0110} (2)_{i}^{n,gh}{\Greekmath 0110} (2)_{i}^{n,ab}|\mathcal{F}_{i-1}^{n}- \overline{C}_{i-1}^{n,gh,ab}\Delta _{n})|\leq K\Delta _{n}^{3/2}(\sqrt{ \Delta _{n}}+{\Greekmath 0111} _{i}^{n}). \end{align} \textbf{Case} $(u,v)\in \{(1,2),(2,1)\}$. By equation ((ref)) we have \begin{equation*} \mathbb{E}(|\widehat{E}(1)|)\leq K\frac{T}{\Delta _{n}}\frac{1}{\Delta _{n}^{1/4}}(\Delta _{n}^{1/4}\widetilde{{\Greekmath 0115} }_{u,v}^{n}\Delta _{n})\leq K\Delta _{n}^{1/2}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{so}\widehat{E}(1)\overset{ \mathbb{P}}{\Rightarrow }0. \end{equation*} \textbf{Case} $(u,v)\in \{(1,1),(2,2)\}$. Set \begin{align*} \widehat{E}^{\prime }(1)& =\frac{1}{\Delta _{n}^{1/4}}\Bigg[ \sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\Theta (u,v)_{0}^{(C),i,n}V_{i-2k_{n}}^{n}\Bigg] \\ \widehat{E}^{\prime \prime }(1)& =\frac{1}{\Delta _{n}^{1/4}}\Bigg[ \sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\Theta (u,v)_{0}^{(C),i,n}\big( V_{i-1}^{n}-V_{i-2k_{n}}^{n}\big)\Bigg] \\ \widehat{E}^{\prime \prime \prime }(1)& =\frac{1}{\Delta _{n}^{1/4}}\Bigg[ \sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\Theta (u,v)_{0}^{(C),i,n}\Big( \mathbb{E}({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|\mathcal{F} _{i-1}^{n})-V_{i-1}^{n}\Big)\Bigg] \end{align*} where \begin{equation*} V_{i-1}^{n}= \begin{cases} C_{i-1}^{n,ga}C_{i-1}^{n,hb}+C_{i-1}^{n,gb}C_{i-1}^{n,ha} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \\ \overline{C}_{i-1}^{n,gh,ab}\Delta _{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,1) \\ 0 & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise} \end{cases} \end{equation*} Note that we have $\widehat{E}(1)=\widehat{E}^{\prime }(1)+\widehat{E} ^{\prime \prime }(1)+\widehat{E}^{\prime \prime \prime }(1)$. Using equations ((ref)) and ((ref)), it can be shown that \begin{equation*} \mathbb{E}(|\widehat{E}^{\prime \prime \prime }(1)|)\leq \begin{cases} K\frac{1}{\Delta _{n}^{5/4}}(\Delta _{n}^{1/4}\widetilde{{\Greekmath 0115} } _{u,v}^{n})\Delta _{n}^{1/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,1) \\ K\frac{1}{\Delta _{n}^{5/4}}(\Delta _{n}^{1/4}\widetilde{{\Greekmath 0115} } _{u,v}^{n})\Delta _{n}^{3/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \end{cases} \leq K\Delta _{n}^{1/2}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{in all cases.} \end{equation*} Next, we prove $\widehat{E}^{\prime }(1)\overset{\mathbb{P}}{\Rightarrow }0$. To this end, write \begin{equation*} \widehat{E}^{\prime }(1)=\frac{1}{\Delta _{n}^{1/4}}\Bigg[ \sum_{i=1}^{[T/\Delta _{n}]-2k_{n}+1}\Theta (u,v)_{0}^{(C),i-1+2k_{n},n}V_{i-1}^{n}\Bigg]. \end{equation*} Using the $\mathcal{F}_{i+2k_{n}-2}^{n}$-measurability of the last sum, we are able to show \begin{align*} \frac{1}{\Delta _{n}^{1/4}}\Bigg[\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}| \mathbb{E}(\Theta (u,v)_{0}^{(C),i-1+2k_{n},n}V_{i-1}^{n}|\mathcal{F} _{i-1}^{n})|\Bigg]& \overset{\mathbb{P}}{\Rightarrow }0\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \\ \frac{2k_{n}-2}{\Delta _{n}^{1/2}}\Bigg[\sum_{i=k_{n}+1}^{[T/\Delta _{n}]-3k_{n}+1}\mathbb{E}\Big(|\Theta (u,v)_{0}^{(C),i-1+2k_{n},n}V_{i-1}^{n})|^{2}\Big)\Bigg]& \Rightarrow 0. \end{align*} The first result readily follows from the inequality \begin{equation*} |\mathbb{E}(\Theta (u,v)_{0}^{(C),i-1+2k_{n},n}V_{i-1}^{n}|\mathcal{F} _{i-1}^{n})|\leq \begin{cases} K\Delta _{n}^{1/2}\widetilde{{\Greekmath 0115} }_{u,v}^{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} (u,v)=(1,1) \\ K\Delta _{n}^{1/2}\widetilde{{\Greekmath 0115} }_{u,v}^{n}\Delta _{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} (u,v)=(2,2) \end{cases} \leq K\Delta _{n}^{3/2}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{in all cases,} \end{equation*} while the second is a direct consequence of \begin{equation*} \mathbb{E}(|\Theta (u,v)_{0}^{(C),i-1+2k_{n},n}V_{i-1}^{n}|^{2})\leq \begin{cases} K\Delta _{n}^{1/2}(\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(1,1) \\ K\Delta _{n}^{1/2}(\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{2}\Delta _{n}^{2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \end{cases} \leq K\Delta _{n}^{5/2}\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{in all cases.} \end{equation*} Finally, to prove that $\widehat{E}^{\prime \prime }(1)\overset{\mathbb{P}}{ \Longrightarrow }0$, we use the fact that \begin{align*} \mathbb{E}(|\Theta (u,v)_{0}^{(C),i,n}\big(V_{i-1}^{n}-V_{i-2k_{n}}^{n}\big) |)& \leq \mathbb{E}(|\Theta (u,v)_{0}^{(C),i,n}|^{2})^{1/2}\mathbb{E} (|V_{i-1}^{n}-V_{i-2k_{n}}^{n}|^{2})^{1/2} \\ & \leq \begin{cases} K\Delta _{n}^{1/2}\widetilde{{\Greekmath 0115} }_{u,v}^{n} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if} (u,v)=(1,1) \\ K\Delta _{n}^{1/4}\widetilde{{\Greekmath 0115} }_{u,v}^{n}\Delta _{n}\Delta _{n}^{1/4} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}(u,v)=(2,2) \end{cases} , \end{align*} which follows from the Cauchy-Schwartz inequality and earlier bounds. In particular, successive conditioning together with Assumption (ref) imply that for $(u,v)=(1,1)$ and $(2,2)$,\newline $\mathbb{E}(|V_{i-1}^{n}-V_{i-2k_{n}}^{n}|^{2})\leq \Delta _{n}^{1/2}$. \subsubsection{Proof of Equation ((ref)) for $w=2$} Our aim here is to show that \begin{align*} & \widehat{E}(2)=\frac{1}{\Delta _{n}^{1/4}}\sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}\Bigg(\sum_{m=1}^{2k_{n}-1}\Big(\frac{3}{2k_{n}^{3}} \sum_{j=0}^{2k_{n}-m-1}\big[(\partial _{gh}H\partial _{ab}G)(C_{i-j-m-1}^{n})-(\partial _{gh}H\partial _{ab}G)(C_{i-2k_{n}}^{n}) \big]{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n}\Big)\times \\ & {\Greekmath 0110} (u)_{i-m}^{n,gh}\Bigg){\Greekmath 0110} (v)_{i}^{n,ab}\overset{\mathbb{P}}{ \Longrightarrow }0. \end{align*} For this purpose, we introduce some new notation. For any $0\leq m\leq 2k_{n}-1$, set \begin{align*} & \Theta (u,v)_{m}^{(C),i,n}=\frac{3}{2k_{n}^{3}}\sum_{j=0}^{2k_{n}-m-1}\big[ (\partial _{gh}H\partial _{ab}G)(C_{i-j-m-1}^{n})-(\partial _{gh}H\partial _{ab}G)(C_{i-2k_{n}}^{n})\big]{\Greekmath 0122} (u)_{j}^{n}{\Greekmath 0122} (v)_{j+m}^{n} \\ & {\Greekmath 011A} (u,v)^{(C),i,n,gh}=\sum_{m=1}^{2k_{n}-1}\Theta (u,v)_{m}^{(C),i,n}{\Greekmath 0110} (u)_{i-m}^{n,gh}. \end{align*} It is easy to see that $\Theta (u,v)_{m}^{(C),i,n}$ is $\mathcal{F} _{i-m-1}^{n}$ measurable and satisfies, by H\"{o}lder inequality, \begin{equation*} |\Theta (u,v)_{m}^{(C),i,n}|\leq \widetilde{{\Greekmath 0115} }_{u,v}^{n} \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\mathbb{E}\Big(|\Theta (u,v)_{m}^{(C),i,n}|^{q}\big| \mathcal{F}_{i-2k_{n}}^{n}\Big)\leq K_{q}\Delta _{n}^{q/4}(\widetilde{ {\Greekmath 0115} }_{u,v}^{n})^{q}. \end{equation*} Lemma (ref) implies that for $q\geq 2$, \begin{equation} \mathbb{E}(|{\Greekmath 011A} (u,v)^{(C),i,n,gh}|^{q})\leq \begin{cases} K_{q}(\Delta _{n}^{1/4}\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{q}k_{n}^{q/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=1 \\ K_{q}(\Delta _{n}^{1/4}\widetilde{{\Greekmath 0115} }_{u,v}^{n})^{q}/k_{n}^{q/2} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}u=2 \end{cases} \leq \begin{cases} K_{q}/k_{n}^{2q} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}v=1 \\ K_{q}k_{n}^{q} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}v=2 \end{cases} . \end{equation} Set \begin{align*} & \widehat{E}^{\prime }(2)=\frac{1}{\Delta _{n}^{1/4}}\sum_{i=3k_{n}}^{[T/ \Delta _{n}]-k_{n}}{\Greekmath 011A} (u,v)^{(C),i,n,gh}\mathbb{E}({\Greekmath 0110} (v)_{i}^{n,ab}| \mathcal{F}_{i-1}^{n}), \\ & \widehat{E}^{\prime \prime }(2)=\frac{1}{\Delta _{n}^{1/4}} \sum_{i=3k_{n}}^{[T/\Delta _{n}]-k_{n}}{\Greekmath 011A} (u,v)^{(C),i,n,gh}({\Greekmath 0110} (v)_{i}^{n,ab}-\mathbb{E}({\Greekmath 0110} (v)_{i}^{n,ab}|\mathcal{F}_{i-1}^{n})). \end{align*} The martingale increments property implies $\mathbb{E}(|\widehat{E}^{\prime \prime }(2)|^{2})\leq K\Delta _{n}^{1/2}$ in all the cases, which in turn implies $\widehat{E}^{\prime \prime }(2)\overset{\mathbb{P}}{\Longrightarrow }0$. Next, using the bounds on ${\Greekmath 011A} (u,v)^{(C),i,n,gh}$, we obtain that $ \widehat{E}^{\prime }(2)\overset{\mathbb{P}}{\Longrightarrow }0$. We refer to jacodrosenbaum-sqrtn for the proofs of Lemma (ref) and Lemma (ref). \subsection{Proof of Lemma (ref)} Set \begin{align*} & {\Greekmath 0118}_i^n={\Greekmath 0127}^n_{i-1}{\Greekmath 0110}_i^n, {\Greekmath 0118}_i^{^{\prime }n}=\mathbb{E}({\Greekmath 0118}_i|\mathcal{F}_{i-1}^n)=\mathbb{E} ({\Greekmath 0127}^n_{i-1}{\Greekmath 0110}_i^n|\mathcal{F}_{i-1}^n)={\Greekmath 0127}^n_{i-1}\mathbb{E} ({\Greekmath 0110}_i^n|\mathcal{F}_{i-1}^n),\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} {\Greekmath 0118}_i^{^{\prime \prime }n}={\Greekmath 0118}_i^{n}-{\Greekmath 0118}_i^{^{\prime }n}. \end{align*} Given that $\|\mathbb{E}({\Greekmath 0110}_{i}^n|\mathcal{F}_{i-1}^n)\| \leq L^{\prime}$ , we have $\|{\Greekmath 0118}_i^{^{\prime }n}\|\leq L^{\prime} |{\Greekmath 0127}^n_{i-1}|$. By the convexity of the function $x^q$, which holds for $q\geq 2$, we have \begin{align*} \|\sum_{j=1}^{2k_n-1}{\Greekmath 0118}_{i+j}^{n}\|^q \leq K\Big(\|\sum_{j=1}^{2k_n-1} {\Greekmath 0118}_{i+j}^{^{\prime }n}\|^q+\|\sum_{j=1}^{2k_n-1}{\Greekmath 0118}_{i+j}^{^{\prime \prime }n}\|^q\Big). \end{align*} Therefore, on the one hand we have \begin{align*} &\|\sum_{j=1}^{2k_n-1}{\Greekmath 0118}_{i+j}^{^{\prime }n}\|^q \leq Kk_n^{q-1} \sum_{j=1}^{2k_n-1}\|{\Greekmath 0118}_{i+j}^{^{\prime }n}\|^q \leq Kk_n^{q-1}L^{\prime q }\sum_{j=1}^{2k_n-1} |{\Greekmath 0127}_{i+j-1}^n|^q, \end{align*} which by $\mathbb{E}\Big(\|{\Greekmath 0127}_{i+j-1}^n\|^q \Big|\mathcal{F}_{i-1}^n \Big) \leq L^q$, satisfies \begin{align*} &\mathbb{E}(\|\sum_{j=1}^{2k_n-1}{\Greekmath 0118}_{i+j}^{^{\prime }n}\|^q | \mathcal{F} _{i-1}^n)\leq KL^{\prime q}k_n^{q-1} \sum_{j=1}^{2k_n-1} \mathbb{E} (|{\Greekmath 0127}_{i+j-1}^n|^q|\mathcal{F}_{i-1}^n)\leq KL^{\prime q}k_n^qL^q. \end{align*} On the other hand, we have $\mathbb{E}(\|{\Greekmath 0118}_{i+j}^{^{\prime \prime }n}\|^q | \mathcal{F}_{i-1}^n)\leq \mathbb{E}(\|{\Greekmath 0118}_{i+j}^{n}\|^q | \mathcal{F} _{i-1}^n) \leq L_qL^q$ and $\mathbb{E}({\Greekmath 0118}_{i+j}^{^{\prime \prime }n} | \mathcal{F}_{i-1}^n)=0$, where the first inequality is a consequence of $ \mathbb{E}(\|{\Greekmath 0118}_{i+j}^{^{\prime }n}\|^q | \mathcal{F}_{i-1}^n)\leq \mathbb{E }(\|{\Greekmath 0118}_{i+j}^{n}\|^q | \mathcal{F}_{i-1}^n) \leq L_qL^q$, which follows from the Jensen's inequality and the law of iterated expectations. Hence, by Lemma B.2 of yacjacod14 we have \begin{align*} \mathbb{E}(\|\sum_{j=1}^{2k_n-1}{\Greekmath 0118}_{i+j}^{^{\prime \prime }n}\|^q | \mathcal{F}_{i-1}^n) \leq K_qL^qL_qk_n^{q/2}. \end{align*} To see the latter, we first prove that the required condition $\mathbb{E} (\|{\Greekmath 0118}_{i}^{n}\|^q | \mathcal{F}_{i-1}^n) \leq L_qL^q$) in the Lemma B.2 of yacjacod14 can be replaced by $\mathbb{E}(\|{\Greekmath 0118}_{i+j}^{n}\|^q | \mathcal{F}_{i-1}^n) \leq L_qL^q$) for $1\leq j \leq 2k_n-1$ without altering the result.\newline \subsection{Proof of Lemma (ref)} We use $i\in L\left( n,T\right) $ throughout the proof of Lemma (ref). We use the terminology \textquotedblleft successive conditioning" to refer to either of the following two equalities, \begin{eqnarray*} x_{1}y_{1}-x_{0}y_{0} &=&x_{0}(y_{1}-y_{0})+y_{0}(x_{1}-x_{0})+(x_{1}-x_{0})(y_{1}-y_{0}), \\ x_{1}y_{1}z_{1}-x_{0}y_{0}z_{0} &=&x_{0}y_{0}(z_{1}-z_{0})+x_{0}z_{0}(y_{1}-y_{0})+y_{0}z_{0}(x_{1}-x_{0})+x_{0}(y_{0}-y_{1})(z_{0}-z_{1}) \\ &&+y_{0}(x_{0}-x_{1})(z_{0}-z_{1})+z_{0}(x_{0}-x_{1})(y_{0}-y_{1})+(x_{1}-x_{0})(y_{1}-y_{0})(z_{1}-z_{0}), \end{eqnarray*} which hold for any real numbers $x_{0},y_{0},z_{0},x_{1},y_{1},$ and $z_{1}$ . \newline To prove Lemma (ref), we first note that ${\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}$ is $\mathcal{F}_{i+2k_{n}}^{n}$-measurable. Therefore, by the law of iterated expectations, we have \begin{equation*} \mathbb{E}\Big({\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}{\Greekmath 0115} _{i+2k_{n}}^{n,gh}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}|\mathcal{F}_{i}^{n}\Big)= \mathbb{E}\Big({\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}\mathbb{E}\big({\Greekmath 0115} _{i+2k_{n}}^{n,gh}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}|\mathcal{F}_{i+2k_{n}}^{n}\big)| \mathcal{F}_{i}^{n}\Big). \end{equation*} By equation (3.27) in jacodrosenbaum-sqrtn, we have \begin{align*} & |\mathbb{E}({\Greekmath 0115} _{i+2k_{n}}^{n,gh}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}|\mathcal{F }_{i+2k_{n}}^{n})-\frac{2}{k_{n}} (C_{i+2k_{n}}^{n,ga}C_{i+2k_{n}}^{n,hb}+C_{i+2k_{n}}^{n,gb}C_{i+2k_{n}}^{n,ha})- \frac{2k_{n}\Delta _{n}}{3}\overline{C}_{i+2k_{n}}^{n,gh,ab}| \\ & \leq K\sqrt{\Delta _{n}}(\Delta _{n}^{1/8}+{\Greekmath 0111} _{i+2k_{n},2k_{n}}^{n}), \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \\ & |\mathbb{E}({\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}|\mathcal{F}_{i}^{n})- \frac{2}{k_{n}}(C_{i}^{n,jl}C_{i}^{n,km}+C_{i}^{n,jm}C_{i}^{n,kl})-\frac{ 2k_{n}\Delta _{n}}{3}\overline{C}_{i}^{n,jk,lm}|\leq K\sqrt{\Delta _{n}} (\Delta _{n}^{1/8}+{\Greekmath 0111} _{i,2k_{n}}^{n}). \end{align*} From the above, it follows that \begin{align*} & |\mathbb{E}\Big({\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}\Big[\mathbb{E} ({\Greekmath 0115} _{i+2k_{n}}^{n,gh}{\Greekmath 0115} _{i+2k_{n}}^{n,ab}\Big|\mathcal{F} _{i+2k_{n}}^{n})-\frac{2}{k_{n}} (C_{i+2k_{n}}^{n,ga}C_{i+2k_{n}}^{n,hb}+C_{i+2k_{n}}^{n,gb}C_{i+2k_{n}}^{n,ha})- \frac{2k_{n}\Delta _{n}}{3}\overline{C}_{i+2k_{n}}^{n,gh,ab}\Big]\Bigg| \mathcal{F}_{i}^{n}\Big)| \\ & \leq \sqrt{\Delta _{n}}\mathbb{E}(|{\Greekmath 0115} _{i}^{n,jk}||{\Greekmath 0115} _{i}^{n,lm}|(\Delta _{n}^{1/8}+{\Greekmath 0111} _{i+2k_{n},2k_{n}}^{n})|\Big|\mathcal{F}_{i}^{n})\leq K\sqrt{\Delta _{n}} \Delta _{n}^{1/8}\mathbb{E}(|{\Greekmath 0115} _{i}^{n,jk}||{\Greekmath 0115} _{i}^{n,lm}|\Big| \mathcal{F}_{i}^{n}) \\ & +K\sqrt{\Delta _{n}}\mathbb{E}(|{\Greekmath 0115} _{i}^{n,jk}||{\Greekmath 0115} _{i}^{n,lm}|{\Greekmath 0111} _{i+2k_{n},2k_{n}}^{n}|\Big|\mathcal{F}_{i}^{n})\leq K\Delta _{n}(\Delta _{n}^{1/8}+{\Greekmath 0111} _{i,4k_{n}}^{n}), \end{align*} where the last inequality follows from Lemma (ref). \newline Now, using equation ((ref)) successively with $Z=C$ and $Z= \overline{C}$ (recall that the latter holds under Assumption (ref) ), together with the successive conditioning, we also have \begin{align*} & |\mathbb{E}\Big({\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}\Big[\frac{2}{k_{n}} (C_{i+2k_{n}}^{n,ga}C_{i+2k_{n}}^{n,hb}+C_{i+2k_{n}}^{n,gb}C_{i+2k_{n}}^{n,ha})+ \frac{2k_{n}\Delta _{n}}{3}\overline{C}_{i+2k_{n}}^{n,gh,ab}-\frac{2}{k_{n}} (C_{i}^{n,ga}C_{i}^{n,hb}+C_{i}^{n,gb}C_{i}^{n,ha}) \\ & -\frac{2k_{n}\Delta _{n}}{3}\overline{C}_{i}^{n,gh,ab}\Big] \Big|\mathcal{F}_{i}^{n}\Big)|\leq K\Delta _{n}\Delta _{n}^{1/4}, \\ & |\mathbb{E}\Big({\Greekmath 0115} _{i}^{n,jk}{\Greekmath 0115} _{i}^{n,lm}\Big[\frac{2}{k_{n}} (C_{i}^{n,ga}C_{i}^{n,hb}+C_{i}^{n,gb}C_{i}^{n,ha}) \\ & +\frac{2k_{n}\Delta _{n}}{3}\overline{C}_{i}^{n,gh,ab}\Big]- \Big[\frac{2}{k_{n}}(C_{i}^{n,jl}C_{i}^{n,km}+C_{i}^{n,jm}C_{i}^{n,kl})+ \frac{2k_{n}\Delta _{n}}{3}\overline{C}_{i}^{n,jk,lm}\Big] \\ & \times \Big[\frac{2}{k_{n}} (C_{i}^{n,ga}C_{i}^{n,hb}+C_{i}^{n,gb}C_{i}^{n,ha})+\frac{2k_{n}\Delta _{n}}{ 3}\overline{C}_{i}^{n,gh,ab}\Big]\Big|\mathcal{F}_{i}^{n}\Big)|\leq K\Delta _{n}(\Delta _{n}^{1/8}+{\Greekmath 0111} _{i,2k_{n}}^{n}). \end{align*} The result derives from the last inequality.\newline \subsection{Proof of Lemma (ref)} \subsubsection{Proof of Equation ((ref)) in Lemma (ref)} We start by obtaining some useful bounds for some important quantities. First, using the second statement in Lemma (ref) applied to $ Z=Y^{\prime }$, we have \begin{equation} |\mathbb{E}({\Greekmath 010B}_i^{n,jk}|\mathcal{F}_{i-1}^n)|\leq K\Delta_n^{3/2}(\sqrt{ \Delta_n}+{\Greekmath 0111}^n_{i,1}). \end{equation} Second, by repeated application of the Cauchy-Schwartz inequality and making use of the third and last statements in Lemma (ref) as well as equation ((ref)) with $Z=C$, it can be shown that \begin{align} \Big|\mathbb{E}({\Greekmath 010B}_i^{n,jk}{\Greekmath 010B}_i^{n,lm}&|\mathcal{F}_{i-1}^n)-\Delta_n^2 \Big(C_i^{n,jl}C_i^{n,km}+C_i^{n,jm}C_i^{n,kl}\Big)\Big|\leq K\Delta_n^{5/2}. \end{align} Next, by successive conditioning and using the bound in equation ((ref)) for $Z=C$ as well as equations ((ref)) and ((ref)), we have for $0\leq u \leq k_n-1$, \begin{align} \Big|\mathbb{E}({\Greekmath 010B}_{i+u}^{n,jk}\big|\mathcal{F}_{i-1}^n)\Big|\leq K\Delta_n^{3/2}(\sqrt{\Delta_n}+{\Greekmath 0111}^n_{i,u}), \end{align} \begin{align} \Big|\mathbb{E}({\Greekmath 010B}_{i+u}^{n,jk}{\Greekmath 010B}_{i+u}^{n,lm}&|\mathcal{F} _{i-1}^n)-\Delta_n^2\Big(C_i^{n,jl}C_i^{n,km}+C_i^{n,jm}C_i^{n,kl}\Big)\Big|\leq K\Delta_n^{5/2}. \end{align} To prove equation ((ref)), we first observe that $ {\Greekmath 0117}_i^{n,jk}{\Greekmath 0117}_i^{n,lm}{\Greekmath 0117}_i^{n,gh}$ can be decomposed as \begin{align*} &{\Greekmath 0117}_i^{n,jk}{\Greekmath 0117}_i^{n,lm}{\Greekmath 0117}_i^{n,gh}=\frac{1}{k_n^3\Delta_n^3} \sum_{u=0}^{k_n-1}{\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,u}^{n,gh}+ \frac{1}{k_n^3\Delta_n^3}\sum_{u=0}^{k_n-2}\sum_{v=u+1}^{k_n-1}\Big[ {\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,v}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh}+ {\Greekmath 0110}_{i,u}^{n,gh}{\Greekmath 0110}_{i,v}^{n,jk}{\Greekmath 0110}_{i,v}^{n,lm} \\ & +{\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh}{\Greekmath 0110}_{i,v}^{n,jk}\Big]+ \frac{1}{k_n^3\Delta_n^3}\sum_{u=0}^{k_n-2}\sum_{v=u+1}^{k_n-1}[ {\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh}+ {\Greekmath 0110}_{i,u}^{n,gh}{\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,v}^{n,lm}+{\Greekmath 0110}_{i,u}^{n,lm} {\Greekmath 0110}_{i,u}^{n,gh}{\Greekmath 0110}_{i,v}^{n,jk}\Big] \\ & +\frac{1}{k_n^3\Delta_n^3}\sum_{u=0}^{k_n-3} \sum_{v=u+1}^{k_n-2}\sum_{w=v+1}^{k_n-1}\Big[{\Greekmath 0110}_{i,u}^{n,jk} {\Greekmath 0110}_{i,v}^{n,lm}{\Greekmath 0110}_{i,w}^{n,gh}+{\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,v}^{n,gh} {\Greekmath 0110}_{i,w}^{n,lm}+{\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,v}^{n,jk}{\Greekmath 0110}_{i,w}^{n,gh}+ {\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh}{\Greekmath 0110}_{i,w}^{n,jk} \\ & +{\Greekmath 0110}_{i,u}^{n,gh}{\Greekmath 0110}_{i,v}^{n,lm}{\Greekmath 0110}_{i,w}^{n,jk}+ {\Greekmath 0110}_{i,u}^{n,gh}{\Greekmath 0110}_{i,v}^{n,jk}{\Greekmath 0110}_{i,w}^{n,lm}\Big], \end{align*} with ${\Greekmath 0110}_{i,u}^{n}={\Greekmath 010B}_{i+u}^{n}+(C_{i+u}^{n}-C_i^{n})\Delta_n$, which satisfies $\mathbb{E}(\|{\Greekmath 0110}_{i,u}^n\|^q|\mathcal{F}_{i-1}^n)\leq K\Delta_n^q$ for $q\geq 2$.\newline Set \begin{align*} &{\Greekmath 0118}_i^n(1)=\frac{1}{k_n^3\Delta_n^3}\sum_{u=0}^{k_n-1}{\Greekmath 0110}_{i,u}^{n,jk} {\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,u}^{n,gh},{\Greekmath 0118}_i^n(2)=\frac{1}{ k_n^3\Delta_n^3}\sum_{u=0}^{k_n-2}\sum_{v=u+1}^{k_n-1}{\Greekmath 0110}_{i,u}^{n,jk} {\Greekmath 0110}_{i,v}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh} \\ &{\Greekmath 0118}_i^n(3)=\frac{1}{k_n^3\Delta_n^3}\sum_{u=0}^{k_n-2}\sum_{v=u+1}^{k_n-1} {\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,u}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh} \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} {\Greekmath 0118}_i^n(4)=\frac{1}{k_n^3\Delta_n^3}\sum_{u=0}^{k_n-3}\sum_{v=u+1}^{k_n-2} \sum_{w=v+1}^{k_n-1}{\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,v}^{n,lm}{\Greekmath 0110}_{i,w}^{n,gh}. \end{align*} The following bounds complete the proof of equation ((ref)), \begin{align} & |\mathbb{E}({\Greekmath 0118}_i^n(1)|\mathcal{F}_{i-1}^n) |\leq K\Delta_n \\ & |\mathbb{E}({\Greekmath 0118}_i^n(2)|\mathcal{F}_{i-1}^n) |\leq K\Delta_n \\ &|\mathbb{E}({\Greekmath 0118}_i^n(3)|\mathcal{F}_{i-1}^n) |\leq K\Delta_n \\ & |\mathbb{E}({\Greekmath 0118}_i^n(4)|\mathcal{F}_{i-1}^n) |\leq K\Delta_n^{3/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}) . \end{align} These bounds are proved below. {\textbf{Proof of Equation ((ref))}} The result readily follows from an application of the Cauchy Schwartz inequality coupled with the bound $\mathbb{E}(\|{\Greekmath 0110}^n_{i+u}\|^q|\mathcal{F} _{i-1}^n)\leq K_q\Delta_n^q$ for $q\geq 2$. {\textbf{Proof of Equation ((ref))}} Using the law of iterated expectation, we have, for $u<v$, \begin{equation} \mathbb{E}({\Greekmath 0110}_{i+u}^{n,jk}{\Greekmath 0110}_{i+v}^{n,lm}{\Greekmath 0110}_{i+v}^{n,gh}|\mathcal{F} _{i-1}^n)=\mathbb{E}({\Greekmath 0110}_{i+u}^{n,jk}\mathbb{E}({\Greekmath 0110}_{i+v}^{n,lm} {\Greekmath 0110}_{i+v}^{n,gh}|\mathcal{F}_{i+u+1}^n)\big|\mathcal{F}_{i-1}^n). \end{equation} By successive conditioning, equation ((ref)), and the Cauchy-Schwartz inequality, we also have \begin{equation*} |\mathbb{E}({\Greekmath 0110}_{i,v}^{n,lm}{\Greekmath 0110}_{i,v}^{n,gh}|\mathcal{F} _{i+u}^n)- \Delta_n^2(C_{i+u+1}^{n,lg}C_{i+u+1}^{n,mh}+C_{i+u+1}^{n,lh}C_{i+u+1}^{n,mg}) \\ -\Delta_n^2(C_{i+u+1}^{n,gh}-C_{i}^{n,gh})(C_{i+u+1}^{n,lm}-C_{i}^{n,lm})| \leq K\Delta_n^{5/2}. \end{equation*} Given that $\mathbb{E}(|{\Greekmath 0110}_{i+u}^{n,jk}|^q\big|\mathcal{F}_{i-1}^n)\leq \Delta^q_n$, the approximation error involved in replacing $\mathbb{E} ({\Greekmath 0110}_{i+v}^{n,lm}{\Greekmath 0110}_{i+v}^{n,gh}|\mathcal{F}_{i+u+1}^n)$ by\newline $ \Delta_n^2(C_{i+u+1}^{n,lg}C_{i+u+1}^{n,mh}+C_{i+u+1}^{n,lh}C_{i+u+1}^{n,mg})+\Delta_n^2(C_{i+u+1}^{n,gh}-C_{i}^{n,gh})(C_{i+u+1}^{n,lm}-C_{i}^{n,lm}) $ in equation ((ref)) is smaller than $\Delta_n^{7/2}$. \newline We can also easily show that \begin{equation} |\mathbb{E}({\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,lm}-C_{i+u}^{n,lm})|\mathcal{F} _{i-1}^n)|\leq K\Delta_n^{3/2}(\sqrt{\Delta_n}+{\Greekmath 0111}_{i,k_n}^n). \end{equation} Since $(C_{i+u}^n-C_i^n)$ is $\mathcal{F}_{i+u}^n$-measurable, we use the successive conditioning, the Cauchy-Schwartz inequality, equation ((ref)), equation ((ref)), and the fifth statement in Lemma (ref) applied to $Z=c$ to obtain \begin{eqnarray} |\mathbb{E} ({\Greekmath 010B}_{i+u}^{n,gh}(C_{i+u}^{n,lm}-C_i^{n,lm})(C_{i+u}^{n,jk}-C_i^{n,jk})| \mathcal{F}_{i-1}^n)| & \leq & K\Delta_n^{5/2} \notag \\ |\mathbb{E}({\Greekmath 010B}_{i+u}^{n,jk} {\Greekmath 010B}_{i+u}^{n,lm}(C_{i+u}^{n,gh}-C_i^{n,gh})|\mathcal{F}_{i-1}^n)| & \leq & K\Delta_n^{5/2} \\ |\mathbb{E}\big( (C_{i+u}^{n,lm}-C_i^{n,lm})(C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+u}^{n,gh}-C_i^{n,gh}) \big)|\mathcal{F}_{i-1}^n)| & \leq & K\Delta_n. \notag \end{eqnarray} The following inequalities can be established using equation ((ref)), the successive conditioning together with equation ((ref)) for $Z=C$, \begin{eqnarray*} \Big|\mathbb{E} ({\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,lg}C_{i+u+1}^{n,mh}+C_{i+u+1}^{n,lh}C_{i+u+1}^{n,mg})| \mathcal{F}_{i-1}^n)\Big| &\leq & K\Delta_n^{3/2} \\ \Big|\mathbb{E}\Big((C_{i+u}^{n,jk}-C_{i}^{n,jk})\big( C_{i+u+1}^{n,lg}C_{i+u+1}^{n,mh}+C_{i+u+1}^{n,lh}C_{i+u+1}^{n,mg}\big)| \mathcal{F}_{i-1}^n\Big)\Big| & \leq & K\Delta_n^{1/2} \\ \Big|\mathbb{E} ({\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,gh}-C_{i}^{n,gh})(C_{i+u+1}^{n,lm}-C_{i}^{n,lm})| \mathcal{F}_{i-1}^n)\Big| & \leq & K\Delta_n^{3/2}(\sqrt{\Delta_n} +{\Greekmath 0111}_{i,k_n}^n). \end{eqnarray*} The last three inequalities together yield $|\mathbb{E}({\Greekmath 0118}_i^n(2)|\mathcal{F }_{i-1}^n) |\leq K\Delta_n$. {\textbf{Proof of Equation ((ref))}} First, note that, for $u<v$, we have \begin{equation} \mathbb{E}({\Greekmath 0110} _{i+u}^{n,jk}{\Greekmath 0110} _{i+u}^{n,lm}{\Greekmath 0110} _{i+v}^{n,gh}| \mathcal{F}_{i-1}^{n})=\mathbb{E}({\Greekmath 0110} _{i+u}^{n,jk}{\Greekmath 0110} _{i+u}^{n,lm} \mathbb{E}({\Greekmath 0110} _{i+v}^{n,gh}|\mathcal{F}_{i+u}^{n})\big|\mathcal{F} _{i-1}^{n}). \end{equation} By successive conditioning and equation ((ref)), we have \begin{equation} |\mathbb{E}({\Greekmath 010B} _{i+w}^{n,gh}|\mathcal{F}_{i+v}^{n})|\leq K\Delta _{n}^{3/2}(\sqrt{\Delta _{n}}+{\Greekmath 0111} _{i+v+1,w-v}). \end{equation} Using the first statement of Lemma applied to $Z=C$, it can be shown that \begin{eqnarray*} &&|\mathbb{E}\big((C_{i+w}^{n,gh}-C_{i+v+1}^{n,gh}))|\mathcal{F}_{i-1}^{n}\big) -\Delta _{n}(w-v-1)\widetilde{b}_{i+v+1}^{n,gh}| \\ &\leq &K(w-v-1)\Delta _{n}{\Greekmath 0111} _{i+v+1,w-v}\leq K\Delta _{n}^{1/2}{\Greekmath 0111} _{i+v+1,w-v}. \end{eqnarray*} The last two inequalities together imply \begin{equation} \Big|\mathbb{E}\Big({\Greekmath 0110} _{i+w}^{n,gh}|\mathcal{F}_{i+v}^{n}\Big) -(C_{i+v+1}^{n,gh}-C_{i}^{n,gh})\Delta _{n}-\Delta _{n}^{2}(w-v-1)\widetilde{ b}_{i+v+1}^{n,gh}\Big|\leq K\Delta _{n}^{3/2}(\sqrt{\Delta _{n}}+{\Greekmath 0111} _{i+v+1,w-v}). \end{equation} Since $\mathbb{E}(|{\Greekmath 0110} _{i,u}^{n,jk}|^{q}|\mathcal{F}_{i-1}^{n})\leq \Delta _{n}^{q}$, the error induced by replacing $\mathbb{E}({\Greekmath 0110} _{i+v}^{n,gh}| \mathcal{F}_{i+u}^{n})$ by $(C_{i+v+1}^{n,gh}-C_{i}^{n,gh})\Delta _{n}+\Delta _{n}^{2}(w-v-1)\widetilde{b}_{i+v+1}^{n,gh}$ in equation ((ref)) is smaller that $\Delta _{n}^{7/2}$.\newline Using Cauchy Schwartz inequality, successive conditioning, equation ((ref)), equation ((ref)) for $Z=C$ and the boundedness of $ \widetilde{b}_{t}$ and $C_{t}$ we obtain \begin{eqnarray*} \Big|\mathbb{E}\Big({\Greekmath 010B} _{i+u}^{n,jk}{\Greekmath 010B} _{i+u}^{n,lm}(C_{i+u+1}^{n,jk}-C_{i}^{n,gh})|\mathcal{F}_{i+u-1}^{n}\Big)\Big| &\leq &K\Delta _{n}^{5/2} \\ \Big|\mathbb{E}\Big({\Greekmath 010B} _{i+u}^{n,jk}{\Greekmath 010B} _{i+u}^{n,lm}\widetilde{b} _{i+u+1}^{n,gh}|\mathcal{F}_{i+u-1}^{n}\Big)\Big| &\leq &K\Delta _{n}^{2} \\ \Big|\mathbb{E}\Big({\Greekmath 010B} _{i+u}^{n,jk}(C_{i+u}^{n,lm}-C_{i}^{n,lm})(C_{i+u+1}^{n,gh}-C_{i}^{n,gh})| \mathcal{F}_{i-1}^{n}\Big)\Big| &\leq &K\Delta _{n}^{1/4}\Delta _{n}^{3/2}( \sqrt{\Delta _{n}}+{\Greekmath 0111} _{i,k_{n}}^{n}) \\ \Big|\mathbb{E}\Big({\Greekmath 010B} _{i+u}^{n,jk}(C_{i+u}^{n,lm}-C_{i}^{n,lm}) \widetilde{b}_{i+u+1}^{n,gh}|\mathcal{F}_{i-1}^{n}\Big)\Big| &\leq &\Delta _{n}^{5/4} \\ \Big|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_{i}^{n,gh})(C_{i+u}^{n,lm}-C_{i}^{n,lm})\widetilde{b} _{i+u+1}^{n,gh}|\mathcal{F}_{i-1}^{n}\Big)\Big| &\leq &K\Delta _{n}^{1/2} \\ \Big|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_{i}^{n,jk})(C_{i+u}^{n,lm}-C_{i}^{n,lm})(C_{i+u+1}^{n,gh}-C_{i}^{n,gh})| \mathcal{F}_{i-1}^{n}\Big)\Big| &\leq &K\Delta _{n}. \end{eqnarray*} The above inequalities together yield $|\mathbb{E}({\Greekmath 0118} _{i}^{n}(3)|\mathcal{F }_{i-1}^{n})|\leq K\Delta _{n}$. {\textbf{Proof of Equation ((ref))}} We first observe that ${\Greekmath 0118}_i^n(4)$ can be rewritten as \begin{align*} {\Greekmath 0118}_i^n(4)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1} \sum_{u=0}^{v-1}{\Greekmath 0110}_{i+u}^{n,jk}{\Greekmath 0110}_{i+v}^{n,lm}{\Greekmath 0110}_{i+w}^{n,gh}, \end{align*} where \begin{align*} &{\Greekmath 0110}_{i+u}^{n,jk}{\Greekmath 0110}_{i+v}^{n,lm}{\Greekmath 0110}_{i+w}^{n,gh}=\Bigg[ {\Greekmath 010B}_{i+u}^{n,jk}{\Greekmath 010B}_{i+v}^{n,lm}{\Greekmath 010B}_{i+w}^{n,gh}+ {\Greekmath 010B}_{i+u}^{n,jk}\Delta_n{\Greekmath 010B}_{i+v}^{n,lm}(C_{i+w}^{n,gh}-C_i^{n,gh})+ {\Greekmath 010B}_{i+u}^{n,jk}\Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm}){\Greekmath 010B}_{i+w}^{n,gh} \\ &+\Delta_n^2 {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+v}^{n,lm}-C_i^{n,lm})(C_{i+w}^{n,gh}-C_i^{n,gh})+ \Delta_n(C_{i+u}^{n,jk}-C_i^{n,jk}){\Greekmath 010B}_{i+v}^{n,lm}{\Greekmath 010B}_{i+w}^{n,gh} \\ &+\Delta_n^2(C_{i+u}^{n,jk}-C_i^{n,jk}) {\Greekmath 010B}_{i+v}^{n,lm}(C_{i+w}^{n,gh}-C_i^{n,gh}) +\Delta_n^2(C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+v}^{n,lm}-C_i^{n,lm}) {\Greekmath 010B}_{i+w}^{n,gh} \\ & +\Delta_n^3(C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+v}^{n,lm}-C_i^{n,lm})(C_{i+w}^{n,gh}-C_i^{n,gh}) \Bigg]. \end{align*} Based on the above decomposition, we set \begin{align*} {\Greekmath 0118}_i^n(4)=\sum_{j=1}^8 {\Greekmath 011F}(j), \end{align*} with ${\Greekmath 011F}(j)$ defined below. We aim to show that $|\mathbb{E}({\Greekmath 011F}(j)\big| \mathcal{F}_{i-1}^n)|\leq K\Delta_n^{3/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}^n)$, $ j=1,\ldots,8$.\newline First, set \begin{align*} {\Greekmath 011F}(1)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1} \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}{\Greekmath 010B}_{i+v}^{n,lm}{\Greekmath 010B}_{i+w}^{n,gh}. \end{align*} Upon changing the order of the summation, we have \begin{align*} &{\Greekmath 011F}(1)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm} {\Greekmath 010B}_{i+w}^{n,gh}. \end{align*} Define also \begin{align*} {\Greekmath 011F}^{\prime}(1)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1} \Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm}\mathbb{E} ({\Greekmath 010B}_{i+w}^{n,gh}|\mathcal{F}_{i+v}^n). \end{align*} Note that $\mathbb{E}({\Greekmath 011F}(1)|\mathcal{F}_{i-1}^n)=\mathbb{E}({\Greekmath 011F}^{\prime}(1)| \mathcal{F}_{i-1}^n)$.\newline By Lemma (ref), we have for $q\geq 2$, \begin{align*} &\mathbb{E}\Big(\Big\|\sum_{u=0}^{v-1}{\Greekmath 010B}_{i+u}^{n,jk}\Big\|^q \Big| \mathcal{F}^n_{i-1}\Big)\leq K_q\Delta_n^{3q/4}. \end{align*} The Cauchy-Schwartz inequality yields \begin{align*} &\mathbb{E}\Bigg(\Big|\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm}\mathbb{E} ({\Greekmath 010B}_{i+w}^{n,gh}|\mathcal{F}_{i+v}^n)\Big|\Bigg|\mathcal{F}_{i-1}^n\Bigg) \leq Kk_n^2 \Big[\mathbb{E}\Big(\Big|\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk} \Big|^{4}\Big|\mathcal{F}_{i-1}^n\Big)\Big]^{1/4} \\ &\times\Big[\mathbb{E}\Big(\Big|{\Greekmath 010B}_{i+v}^{n,lm}\Big|^{4}\Big|\mathcal{F} _{i-1}^n\Big)\Big]^{1/4}\times\Big[\mathbb{E}\Big(\Big|\mathbb{E} ({\Greekmath 010B}_{i+w}^{n,gh}|\mathcal{F}_{i+v}^n)\Big|^{2}\Big|\mathcal{F}_{i-1}^n\Big) \Big]^{1/2}\leq K\Delta_n k_n^2 \Delta_n^{3/4} \Delta_n^{3/2}(\sqrt{\Delta_n} +{\Greekmath 0111}_{i,k_n}^{n}), \end{align*} where the last iteration is obtained using equation ((ref)) as well as the inequality $(a+b)^{1/2}\leq a^{1/2}+b^{1/2}$, which holds for positive real numbers $a$ and $b$, and the third statement in Lemma (ref). It follows that \begin{align*} |\mathbb{E}\Big({\Greekmath 011F}(1)\big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n^{3/4} (\sqrt{ \Delta_n}+{\Greekmath 0111}_{i,k_n}^n). \end{align*} Next, we introduce \begin{align*} &{\Greekmath 011F}(2)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1}\Delta_n(C_{i+u}^{n,jk}-C_{i}^{n,jk})\Big) {\Greekmath 010B}_{i+v}^{n,lm}{\Greekmath 010B}_{i+w}^{n,gh}, \\ &{\Greekmath 011F}(3)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+v}^{n,jk}\Big) \Delta_n(C_{i+u}^{n,lm}-C_{i}^{n,lm}){\Greekmath 010B}_{i+w}^{n,gh}, \\ &{\Greekmath 011F}(4)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} \Delta_n(C_{i+u}^{n,jk}-C_{i}^{n,jk})\Big) \Delta_n(C_{i+u}^{n,lm}-C_{i}^{n,lm}){\Greekmath 010B}_{i+w}^{n,gh}. \end{align*} Given that for $q \geq 2$, we have \begin{align*} &\mathbb{E}\Big(\Big\|\sum_{u=0}^{v-1}\Delta_n(C_{i+u}^{n,jk}-C_{i}^{n,jk}) \Big\|^q \Big|\mathcal{F}^n_{i-1}\Big)\leq K_q\Delta_n^{3q/4} \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \mathbb{E}(\|C_{i+u}^{n,jk}-C_{i}^{n,jk}\|^q\big|\mathcal{F}_{i-1}^n)\leq K_q \Delta_n^{q/4}. \end{align*} Similar steps to ${\Greekmath 011F}(1)$ lead to \begin{align*} |\mathbb{E}({\Greekmath 011F}(2)\big|\mathcal{F}_{i-1}^n)|\leq K\Delta_n^{3/4} (\sqrt{\Delta_n }+{\Greekmath 0111}_{i,k_n}^n) \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} |\mathbb{E}({\Greekmath 011F}(j)\big|\mathcal{F}_{i-1}^n)|\leq K\Delta_n(\sqrt{\Delta_n}+{\Greekmath 0111}_{i,k_n}^n) \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for} j=3,4. \end{align*} Define \begin{align*} &{\Greekmath 011F}(5)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm} \Delta_n(C_{i+w}^{n,gh}-C_i^{n,gh}) \\ &{\Greekmath 011F}^{\prime}(5)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1} \Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm}\Delta_n \mathbb{E}\big((C_{i+w}^{n,gh}-C_i^{n,gh})\big|\mathcal{F}_{i+v}^n) \\ &{\Greekmath 011F}(6)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} \Delta_n(C_{i+u}^{n,jk}-C_i^{n,jk})\Big){\Greekmath 010B}_{i+v}^{n,lm} \Delta_n(C_{i+w}^{n,gh}-C_i^{n,gh}) \\ &{\Greekmath 011F}(7)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big)\Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm}) \Delta_n(C_{i+w}^{n,gh}-C_{i}^{n,gh}), \end{align*} where we have $\mathbb{E}({\Greekmath 011F}(5)|\mathcal{F}_{i-1}^n)=\mathbb{E} ({\Greekmath 011F}^{\prime}(5)|\mathcal{F}_{i-1}^n)$. Recalling equation ((ref)), we further decompose ${\Greekmath 011F}^{\prime}(5)$ as, \begin{align*} {\Greekmath 011F}^{\prime}(5)=\sum_{j=1}^{5}{\Greekmath 011F}(5)[j], \end{align*} with \begin{align*} {\Greekmath 011F}{\prime}(5)[1]=&\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\Big(\sum_{u=0}^{v-1}{\Greekmath 010B}_{i+u}^{n,jk}\Big) {\Greekmath 010B}_{i+v}^{n,lm}\Big(\mathbb{E}\Big(C_{i+w}^{n,gh}-C_{i}^{n,gh}|\mathcal{F }_{i+v}^n\Big) \\ &-(C_{i+v+1}^{n,gh}-C_{i}^{n,gh})\Delta_n -\widetilde{b}_{i+v+1}^{n,gh} \Delta_n^2(w-v-1)\Big) \\ {\Greekmath 011F}{\prime}(5)[2]=&\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\Delta_n(C_{i+v}^{n,gh}-C_{i}^{n,gh})\Big( \sum_{u=0}^{v-1}{\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm} \\ {\Greekmath 011F}{\prime}(5)[3]=&\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big) \Delta_n(C_{i+v+1}^{n,gh}-C_{i+v}^{n,gh}){\Greekmath 010B}_{i+v}^{n,lm} \\ {\Greekmath 011F}{\prime}(5)[4]=&\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big) \Delta_n^2(w-v-1)(\widetilde{b}_{i+v+1}^{n,gh}-\widetilde{b} _{i+v}^{n,gh}){\Greekmath 010B}_{i+v}^{n,lm} \\ {\Greekmath 011F}{\prime}(5)[5]=&\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\Delta_n^2(w-v-1)\widetilde{b}_{i+v}^{n,gh}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big){\Greekmath 010B}_{i+v}^{n,lm}. \end{align*} Using equations ((ref)), ((ref)), and ((ref)) and following the same strategy proof as for ${\Greekmath 011F}(1)$, it can be shown that \begin{align*} |\mathbb{E}\Big({\Greekmath 011F}{\prime}(5)[j]\big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n^{3/4} (\sqrt{\Delta_n}+{\Greekmath 0111}_{i,k_n}^n), \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for} j=1,\ldots,5, \end{align*} which in turn implies \begin{align*} |\mathbb{E}\Big({\Greekmath 011F}(5)\big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n^{3/4} (\sqrt{ \Delta_n}+{\Greekmath 0111}_{i,k_n}^n), \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for} j=1,\ldots,5. \end{align*} The term ${\Greekmath 011F}(6)$ can be handled similarly to ${\Greekmath 011F}(5)$, hence we conclude that \begin{align*} |\mathbb{E}\Big({\Greekmath 011F}(6)\big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n^{3/4} (\sqrt{ \Delta_n}+{\Greekmath 0111}_{i,k_n}^n). \end{align*} Next, we set \begin{align*} {\Greekmath 011F}(7)=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\Bigg(\sum_{v=0}^{w-1} \Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big) \Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm})\Delta_n(C_{i+w}^{n,gh}-C_{i}^{n,gh}) \Bigg). \end{align*} Define \begin{align*} &{\Greekmath 011F}(7)[1]=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\Bigg(\sum_{v=0}^{w-1} \Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big) \Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm})\Delta_n(C_{i+v+1}^{n,gh}-C_{i+v}^{n,gh}) \Bigg) \\ &{\Greekmath 011F}(7)[2]=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\Bigg(\sum_{v=0}^{w-1} \Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big) \Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm})\Delta_n(C_{i+v}^{n,gh}-C_i^{n,gh})\Bigg) \\ &{\Greekmath 011F}(7)[3]=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\Bigg(\sum_{v=0}^{w-1} \Big(\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big) \Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm})\Delta_n^2(w-v-1)(\widetilde{b} _{i+v+1}^{n,gh}-\widetilde{b}_{i+v}^{n,gh})\Bigg) \\ &{\Greekmath 011F}(7)[4]=\frac{1}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1}\Bigg( \sum_{v=0}^{w-1}\Delta_n^2(w-v-1)\widetilde{b}_{i+v}^{n,gh}\Big( \sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}\Big)\Delta_n(C_{i+v}^{n,lm}-C_i^{n,lm}) \Bigg). \end{align*} It is easy to see that \begin{align*} {\Greekmath 011F}(7)=\sum_{j=1}^4 {\Greekmath 011F}(7)[j]. \end{align*} Similarly to calculations used for ${\Greekmath 011F}(1)$, it can be shown that \begin{align*} |\mathbb{E}({\Greekmath 011F}(7)[j]\big|\mathcal{F}_{i-1}^n)|\leq K\Delta_n^{1/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}), \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for} j=1,\ldots,3. \end{align*} To handle the remaining term ${\Greekmath 011F}(7)[4]$, we decompose it $ {\Greekmath 011F}(7)[4]=\sum_{j=1}^9 {\Greekmath 011F}(7)[4][j]$, where \begin{align*} &{\Greekmath 011F}(7)[4][1]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,lm}-C_{i+u}^{n,lm})(C_{i+u+1}^{n,gh}-C_{i+u}^{n,gh}) \\ &{\Greekmath 011F}(7)[4][2]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,gh}-C_{i}^{n,gh}) {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,lm}-C_{i+u}^{n,lm}) \\ &{\Greekmath 011F}^{\prime}(7)[4][2]=\frac{\Delta_n^2}{(k_n\Delta_n)^3} \sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,gh}-C_{i}^{n,gh})\mathbb{E} ({\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,lm}-C_{i+u}^{n,lm})|\mathcal{F}_{i+u-1}^n) \\ &{\Greekmath 011F}(7)[4][3]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,lm}-C_{i}^{n,lm}) {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,gh}-C_{i+u}^{n,gh}) \\ &{\Greekmath 011F}(7)[4][4]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,lm}-C_{i}^{n,lm})(C_{i+u}^{n,gh}-C_{i}^{n,gh}){\Greekmath 010B}_{i+u}^{n,jk} \\ &{\Greekmath 011F}(7)[4][5]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,lm}-C_{i}^{n,lm}) {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+v}^{n,gh}-C_{i+u+1}^{n,gh}) \\ &{\Greekmath 011F}^{\prime}(7)[2][5]=\frac{\Delta_n^2}{(k_n\Delta_n)^3} \sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,lm}-C_{i}^{n,lm}){\Greekmath 010B}_{i+u}^{n,jk}\mathbb{E} ((C_{i+v}^{n,gh}-C_{i+u+1}^{n,gh}|\mathcal{F}_{i+u-1}^n) \\ &{\Greekmath 011F}(7)[4][6]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,lm}-C_{i+u}^{n,lm})(C_{i+v}^{n,gh}-C_{i+u+1}^{n,gh}) \\ &{\Greekmath 011F}(7)[4][7]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} (C_{i+u}^{n,gh}-C_{i}^{n,gh}) {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+v}^{n,lm}-C_{i+u+1}^{n,lm}) \\ &{\Greekmath 011F}(7)[4][8]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+u+1}^{n,gh}-C_{i+u}^{n,gh})(C_{i+v}^{n,lm}-C_{i+u+1}^{n,lm}) \\ &{\Greekmath 011F}(7)[4][9]=\frac{\Delta_n^2}{(k_n\Delta_n)^3}\sum_{w=2}^{k_n-1} \sum_{v=0}^{w-1}\sum_{u=0}^{v-1} {\Greekmath 010B}_{i+u}^{n,jk}(C_{i+v}^{n,lm}-C_{i+u+1}^{n,lm})(C_{i+v}^{n,gh}-C_{i+u+1}^{n,gh}). \\ \end{align*} Using arguments similar to those involved for the treatment of ${\Greekmath 011F}(1)$, it can be shown that \begin{align*} |\mathbb{E}({\Greekmath 011F}(7)[4][j]\big|\mathcal{F}_{i-1}^n)|\leq K\Delta_n^{1/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}), \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for} j=1,\ldots,8, \end{align*} which yields \begin{align*} |\mathbb{E}({\Greekmath 011F}(7)\big|\mathcal{F}_{i-1}^n)|\leq K\Delta_n^{1/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}). \end{align*} Next, define \begin{align*} {\Greekmath 011F}(8)=\frac{1}{k_n^3}\sum_{w=2}^{k_n-1}\sum_{v=0}^{w-1} \sum_{u=0}^{v-1}(C_{i+u}^{n,jk}-C_{i}^{n,jk})(C_{i+v}^{n,lm}-C_{i}^{n,lm})(C_{i+w}^{n,gh}-C_{i}^{n,gh}). \end{align*} This term can be further decomposed into six components. Successive conditioning and existing bounds give \begin{align*} &|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+v}^{n,lm}-C_{i+u}^{n,lm})(C_{i+w}^{n,gh}-C_{i+v}^{n,gh}) \big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n \\ &|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+v}^{n,lm}-C_{i+u}^{n,lm})(C_{i+v}^{n,gh}-C_{i+u}^{n,gh}) \big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n^{3/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}) \\ &|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+v}^{n,lm}-C_{i+u}^{n,lm})(C_{i+u}^{n,gh}-C_{i}^{n,gh}) \big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n \\ &|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+u}^{n,lm}-C_{i}^{n,lm})(C_{i+w}^{n,gh}-C_{i+v}^{n,gh}) \big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n \\ &|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+u}^{n,lm}-C_{i}^{n,lm})(C_{i+v}^{n,gh}-C_{i+u}^{n,gh}) \big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n \\ &|\mathbb{E}\Big( (C_{i+u}^{n,jk}-C_i^{n,jk})(C_{i+u}^{n,lm}-C_{i}^{n,lm})(C_{i+u}^{n,gh}-C_{i}^{n,gh}) \big|\mathcal{F}_{i-1}^n\Big)|\leq K\Delta_n \end{align*} These bounds can be used to deduce \begin{align*} |\mathbb{E}({\Greekmath 011F}(8)\big|\mathcal{F}_{i-1}^n)|\leq K\Delta_n. \end{align*} This completes the proof. \subsubsection{Proof of Equations ((ref)) and ((ref)) in Lemma (ref)} Observe that \begin{align*} &{\Greekmath 0117}_i^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh})= \frac{1}{k_n\Delta_n}\sum_{u=0}^{k_n-1} {\Greekmath 0110}_{i,u}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}), \\ &{\Greekmath 0117}_i^{n,jk}{\Greekmath 0117}_i^{n,lm}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh})=\frac{1}{ k_n^2\Delta_n^2} \sum_{u=0}^{k_n-1}{\Greekmath 0110}_{i,u}^{n,jk} {\Greekmath 0110}_{i,u}^{n,lm}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) \\ &+\frac{1}{k_n^2\Delta_n^2} \sum_{u=0}^{k_n-2}\sum_{v=0}^{k_n-1} {\Greekmath 0110}_{i,u}^{n,jk}{\Greekmath 0110}_{i,v}^{n,lm}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) +\frac{1 }{k_n^2\Delta_n^2}\sum_{u=0}^{k_n-2}\sum_{v=0}^{k_n-1}{\Greekmath 0110}_{i,u}^{n,lm} {\Greekmath 0110}_{i,v}^{n,jk}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}). \end{align*} Hence, equations ((ref)) and ((ref)) can be proved using the same strategy as for ((ref)). \subsubsection{Proof of Equations ((ref)) and ((ref)) in Lemma (ref)} Note that we have \begin{align*} &{\Greekmath 0115}_i^{n,jk}{\Greekmath 0115}_i^{n,lm}{\Greekmath 0117}_i^{n,gh}={\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk} {\Greekmath 0117}_{i+k_n}^{n,lm}+{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,jk}{\Greekmath 0117}_{i}^{n,lm}-{\Greekmath 0117}_i^{n,gh} {\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk}-{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk} \\ &+{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})-{\Greekmath 0117}_i^{n,gh} {\Greekmath 0117}_{i}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})+{\Greekmath 0117}_i^{n,gh} {\Greekmath 0117}_{i+k_n}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk}) \\ &-{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk}) +{\Greekmath 0117}_i^{n,gh}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,lm}-C_{i}^{n,lm}), \end{align*} and \begin{eqnarray*} &&{\Greekmath 0115}_i^{n,gh}{\Greekmath 0115}_i^{n,jk}{\Greekmath 0115}_i^{n,lm}={\Greekmath 0117}_{i+k_n}^{n,gh} {\Greekmath 0117}_{i+k_n}^{n,jk}{\Greekmath 0117}_{i+k_n}^{n,lm}+{\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i}^{n,jk} {\Greekmath 0117}_{i}^{n,lm}-{\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk}- {\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk} \\ &&+{\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})- {\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})+ {\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk}) \\ &&-{\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk}) +{\Greekmath 0117}_{i+k_n}^{n,gh}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,lm}-C_{i}^{n,lm}) \\ && -{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}{\Greekmath 0117}_{i+k_n}^{n,lm}-{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,jk} {\Greekmath 0117}_{i}^{n,lm} +{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk}+{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,lm} {\Greekmath 0117}_{i+k_n}^{n,jk} \\ &&-{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})+ {\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})-{\Greekmath 0117}_i^{n,gh} {\Greekmath 0117}_{i+k_n}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk}) \\ &&+{\Greekmath 0117}_i^{n,gh}{\Greekmath 0117}_{i}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk}) -{\Greekmath 0117}_i^{n,gh}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,lm}-C_{i}^{n,lm}) \\ && +{\Greekmath 0117}_{i+k_n}^{n,jk}{\Greekmath 0117}_{i+k_n}^{n,lm}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) +{\Greekmath 0117}_{i}^{n,jk}{\Greekmath 0117}_{i}^{n,lm}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) -{\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) \\ && -{\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh})+ {\Greekmath 0117}_{i+k_n}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) \\ &&- {\Greekmath 0117}_{i}^{n,jk}(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) +{\Greekmath 0117}_{i+k_n}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) \\ && -{\Greekmath 0117}_{i}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}) +(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}). \end{eqnarray*} From ((ref)), notice that ${\Greekmath 0117}_i^{n}$ is $\mathcal{F}_{i+k_n-1}^n$ -measurable and satisfies $\|\mathbb{E}({\Greekmath 0117}_i^{n}|\mathcal{F}_{i-1}^n)\|\leq K\Delta_n^{1/2}$.\newline The law of iterated expectations and existing bounds imply \begin{eqnarray} |\mathbb{E}({\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i+k_n}^{n,jk}|\mathcal{F}_{i-1}^n)| & \leq & K\Delta_n^{3/4}, \notag \\ |\mathbb{E}({\Greekmath 0117}_{i}^{n,lm}{\Greekmath 0117}_{i}^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}|\mathcal{F}_{i-1}^n)| & \leq & K\Delta_n, \notag \\ |\mathbb{E}({\Greekmath 0117}_{i}^{n,lm}(C_{i+k_n}^{n,gh}-C_{i}^{n,gh}){\Greekmath 0117}_{i+k_n}^{n,jk}| \mathcal{F}_{i-1}^n)| & \leq & K\Delta_n, \notag \\ |\mathbb{E}({\Greekmath 0117}_{i+k_n}^{n,lm}(C_{i+k_n}^{n,jk}-C_{i}^{n,jk})|\mathcal{F} _{i-1}^n)| & \leq & K\Delta_n^{3/4}, \notag \\ |\mathbb{E} ((C_{i+k_n}^{n,jk}-C_{i}^{n,jk})(C_{i+k_n}^{n,lm}-C_{i}^{n,lm})(C_{i+k_n}^{n,gh}-C_{i}^{n,gh})| \mathcal{F}_{i-1}^n)| & \leq & K\Delta_n. \end{eqnarray} It can also be readily verified that \begin{align*} &|\mathbb{E}({\Greekmath 0117}_{i+k_n}^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,ab}|\mathcal{F}_{i+k_n-1}^n)- \frac{1}{k_n} (C_{i+k_n}^{n,ga}C_{i+k_n}^{n,hb}+C_{i+k_n}^{n,gb}C_{i+k_n}^{n,ha})-\frac{ k_n\Delta_n}{3}\overline{C}_{i+k_n}^{n,gh,ab}| \\ &\leq K\sqrt{\Delta_n}(\Delta_n^{1/8}+{\Greekmath 0111}_{i+k_n,k_n}^n). \end{align*} Hence, for ${\Greekmath 0127}_i^{n,gh} \in\{{\Greekmath 0117}_i^{n,gh},C_{i+k_n}^{n,gh}-C_{i}^{n,gh}\}$, which satisfies $\mathbb{ E}(|{\Greekmath 0127}_i^{n,gh}|^q\Big| \mathcal{F}_{i-1}^n) \leq K\Delta_n^{q/4}$ and $ \mathbb{E}({\Greekmath 0127}_i^{n,gh}|\mathcal{F}_{i-1}^n)\leq K\Delta_n^{1/2}$. One can show that \begin{align*} &|\mathbb{E}({\Greekmath 0127}_i^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}{\Greekmath 0117}_{i+k_n}^{n,lm}|\mathcal{F} _{i-1}^n)-\mathbb{E}\Big({\Greekmath 0127}_i^{n,gh}\Big[\frac{1}{k_n} (C_{i+k_n}^{n,jl}C_{i+k_n}^{n,km}+C_{i+k_n}^{n,jm}C_{i+k_n}^{n,kl})-\frac{ k_n\Delta_n}{3}\overline{C}_{i+k_n}^{n,jk,lm}\Big]|\mathcal{F}_{i-1}^n\Big)| \\ &\leq K\Delta_n^{3/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,2k_n}^n). \end{align*} Next, by combining the successive conditioning together with existing bounds, we have \begin{eqnarray*} |\mathbb{E}({\Greekmath 0127}_i^{n,gh}\overline{C}_{i+k_n}^{n,jk,lm})| & \leq & K\Delta_n^{1/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,k_n}^n) \\ |\mathbb{E}({\Greekmath 0127}_i^{n,gh}C_{i+k_n}^{n,jl}C_{i+k_n}^{n,km})| & \leq & K\Delta_n^{1/2}, \end{eqnarray*} which together imply \begin{equation} |\mathbb{E}({\Greekmath 0127}_i^{n,gh}{\Greekmath 0117}_{i+k_n}^{n,jk}{\Greekmath 0117}_{i+k_n}^{n,lm}|\mathcal{F} _{i-1}^n)|\leq K \Delta_n^{3/4}(\Delta_n^{1/4}+{\Greekmath 0111}_{i,2k_n}^n). \end{equation} It is easy to see that equations ((ref)), ((ref)) and ((ref)) and the inequality ${\Greekmath 0111}_{i,k_n}^n \leq {\Greekmath 0111}_{i,2k_n}^n $ together yield equations ((ref)) and ((ref) ). \subsection{Proof of Lemma (ref)} Equation ((ref)) can be proved easily using the bounds of $ {\Greekmath 011A} (u,v)_{i}^{n,gh}$ in equation (E.60). To show equations ((ref)), ((ref)) and ((ref)), we set \begin{equation*} \overline{\overline{A11}}(H,gh,u;G,ab,v)={\Greekmath 0115} (u,v)_{0}^{n}\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1}){\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}. \end{equation*} Then, \begin{equation*} \frac{1}{\Delta _{n}^{1/4}}\Big(\overline{\overline{A11}}(H,gh,u;G,ab,v)- \overline{A11}(H,gh,u;G,ab,v)\Big)\overset{\mathbb{P}}{\Rightarrow }0. \end{equation*} The above result is proved following similar steps as for equation ((ref)) in case $w=1$ by replacing $\Theta (u,v)_{0}^{(C),i,n}$ by $ {\Greekmath 0115} (u,v)_{0}^{n}((\partial _{gh}H\partial _{ab}G)(C_{i-1})-(\partial _{gh}H\partial _{ab}G)(C_{i-2k_{n}}))$, which has the same bounds as the former. Next, decompose $\overline{\overline{A11}}$ as follows, \begin{align*} \overline{\overline{A11}}(H,gh,u;G,ab,v)& ={\Greekmath 0115} (u,v)_{0}^{n}\Bigg[ \sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1})V_{i-1}^{n} \\ & +\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1})\Big(\mathbb{E}({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}| \mathcal{F}_{i-1}^{n})-V_{i-1}^{n}\Big) \\ & +\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1})\Big({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}-\mathbb{E} ({\Greekmath 0110} (u)_{i}^{n,gh}{\Greekmath 0110} (v)_{i}^{n,ab}|\mathcal{F}_{i-1}^{n})\Big)\Bigg]. \end{align*} We follow the proof of equation ((ref)) for $w=1$, and we replace $\Theta (u,v)_{0}^{(C),i,n}$ by ${\Greekmath 0115} (u,v)_{0}^{n}(\partial _{gh}H\partial _{ab}G)(C_{i-1})$, which satisfies only the condition $ |{\Greekmath 0115} (u,v)_{0}^{n}(\partial _{gh}H\partial _{ab}G)(C_{i-1})|\leq \widetilde{{\Greekmath 0115} }_{u,v}^{n}$. This calculation shows that the last two terms in the above decomposition vanish at a rate faster than $\Delta _{n}^{1/4}$. Therefore, \begin{equation*} \frac{1}{\Delta _{n}^{1/4}}\Bigg(\overline{\overline{A11}} (H,gh,u;G,ab,v)-{\Greekmath 0115} (u,v)_{0}^{n}\Big(\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1})V_{i-1}^{n}\Big)\Bigg) \Rightarrow 0. \end{equation*} As a consequence, for $(u,v)=(1,2)$ and $(2,1)$, \begin{equation*} \frac{1}{\Delta _{n}^{1/4}}\overline{\overline{A11}}(H,gh,u;G,ab,v) \Rightarrow 0. \end{equation*} The results follow from the following observation, \begin{eqnarray*} &&\frac{1}{\Delta _{n}^{1/4}}\Bigg({\Greekmath 0115} (u,v)_{0}^{n}\Big( \sum_{g,h,a,b=1}^{d}\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1})V_{i-1}^{n}(u,v)\Big) \\ &&-\frac{3}{{\Greekmath 0112} ^{2}}\int_{0}^{T}(\partial _{gh}H\partial _{ab}G)(C_{t})(C_{t}^{ga}C_{t}^{hb}+C_{t}^{gb}C_{t}^{ha})dt\Bigg)\Rightarrow 0, \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for }(u,v)=(2,2), \\ &&\frac{1}{\Delta _{n}^{1/4}}\Bigg(\sum_{g,h,a,b=1}^{d}{\Greekmath 0115} (u,v)_{0}^{n} \Big(\sum_{i\in L^{\prime }\left( n,T\right) }(\partial _{gh}H\partial _{ab}G)(C_{i-1})V_{i-1}^{n}(u,v)\Big)-[H(C),G(C)]_{T}\Bigg)\Rightarrow 0, \\ &&\relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for }(u,v)=(1,1). \end{eqnarray*} \captionsetup{font=small,justification=justified} \FloatBarrier \section{Numerical Implementation} We now discuss some details for the numerical implementation of our estimators. Section (ref) explains how the main quantities of interest can be expressed in terms of $\left[ H(C),G(C)\right] _{T}$, where $C$ is the spot variance matrix of all $d$ assets. However, in practice many quantities of interest involve only a much smaller subset of assets, which greatly reduces the computational burden. For example, suppose we want to calculate the variance of the IdioVol for a single stock, where R-FM is the CAPM, and IdioVol-FM has one volatility factor -- the market volatility. Then, we only need to consider two assets, the stock and the market, e.g., SPY, so $d_{S}=d_{F}=1$ and $d=2$. Denote the relevant spot variance-covariance matrix by \begin{equation*} C=\left( \begin{array}{cc} C_{11} & C_{12} \\ C_{21} & C_{22} \end{array} \right) , \end{equation*} where $C_{22}=C_{F}$ is the spot variance of the market, and $C_{11}$ is the spot variance of the individual stock. The quantity of interest is \begin{equation*} \left[ H(C),H(C)\right] _{T}=\left[ C_{Z1},C_{Z1}\right] _{T}, \end{equation*} where $C_{Z1,t}=C_{11}-C_{12}C_{22}^{-1}C_{21}$. The estimators in equations ((ref)) and ((ref)) involve the first derivatives $ \partial _{ab}H\left( C\right) $ for $a,b=1,...,d$, which are \begin{equation*} \partial _{ab}H\left( C\right) \equiv \frac{\partial H\left( C\right) }{ \partial C_{ab}}=\frac{\partial C_{Zj}}{\partial C_{ab}}=\frac{\partial \left( C_{11}-C_{12}C_{22}^{-1}C_{21}\right) }{\partial C_{ab}}=\left\{ \begin{array}{cc} C_{12}C_{22}^{-2}C_{21} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }\left( a,b\right) =\left( 2,2\right) \\ -C_{22}^{-1}C_{21} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }\left( a,b\right) =\left( 1,2\right) \\ -C_{12}C_{22}^{-1} & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }\left( a,b\right) =\left( 2,1\right) \\ 1 & \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }\left( a,b\right) =\left( 1,1\right) \end{array} \right. \end{equation*} If we are interested in the stock's IdioVol ${\Greekmath 010D} _{Z}$, by equation ((ref)) we also need the volatility factor $\Pi _{t}=G\left( C_{t}\right) =C_{22,t}$, and $\left[ \Pi ,C_{Z1}\right] _{T}^{c}$. The derivatives are $\partial _{ab}G\left( C\right) \equiv \partial G\left( C\right) \left/ \partial C_{ab}\right. =1\left\{ a=b=2\right\} $. \section{Additional Figures} \begin{figure}[!h] \caption{Monthly $R^2$ of two Return Factor Models ($\protect\widehat{R} ^2_{Yj}$): the CAPM (the blue dotted line) and the Fama-French three factor model (the red solid line). Stocks are represented by tickers (see Table (ref) for full stock names). } \end{figure} \begin{figure}[!h] \caption{Monthly $R^2$ of two Return Factor Models ($\protect\widehat{R} ^2_{Yj}$): the CAPM (the blue dotted line) and the Fama-French three factor model (the red solid line). Stocks are represented by tickers (see Table (ref) for full stock names). } \end{figure}