A nonparametric test for rough volatility

abstractWe develop a nonparametric test for deciding whether volatility of an asset follows a standard semimartingale process, with paths of finite quadratic variation, or a rough process with paths of infinite quadratic variation. The test utilizes the fact that volatility is rough if and only if volatility increments are negatively autocorrelated at high frequencies. It is based on the sample autocovariance of increments of spot volatility estimates computed from high-frequency asset return data. By showing a feasible CLT for this statistic under the null hypothesis of semimartingale volatility paths, we construct a test with fixed asymptotic size and an asymptotic power equal to one. The test is derived under very general conditions for the data-generating process. In particular, it is robust to jumps with arbitrary activity and to the presence of market microstructure noise. In an application of the test to SPY high-frequency data, we find evidence for rough volatility.

definitionWe say that a function $Z=(Z_t)_{t\in[0,T]}$ is rough if \begin{equation} \frac{\mathit{RV}(Z)^{n,2}_T}{\mathit{RV}(Z)^n_T}= \frac{ \frac12\sum_{i=0}^n \bigl(Z_{\frac{i+1}{n}T}-Z_{\frac{i-1}{n}T}\bigr)^2}{ \sum_{i=1}^n \bigl(Z_{\frac in T}-Z_{\frac{i-1}{n} T}\bigr)^2} \stackrel{p}{\longrightarrow} \tau \quad as n\to\infty, \end{equation} for some $\tau<1$. We say that a sequence $(a_n)_{n\in\mathbb{N}}$ converges subsequentially to a limit $a$, denoted by $a_n\stackrel{p}{\longrightarrow} a$, if for every subsequence $(n_k)_{k\in\mathbb{N}}$ there is another subsequence $(n_{k_\ell})_{\ell\in\mathbb{N}}$ such that $a_{n_{k_\ell}}\to a$ as $\ell\to\infty$. Moreover, in (ref), we use the convention $0/0=2$ and define $Z_t=Z_0$ for $t<0$ and $Z_t=Z_T$ for $t>T$.

remarkThere are other notions of roughness in the literature; see e.g., cont2022rough and HS24. These definitions are based on $p$-variations for different values of $p$ and produce the same ranking as Definition (ref) when used to compare the roughness of continuous processes (for discontinuous processes, there are some differences). The main reason we use Definition (ref) in this paper is because, as we show in Theorems (ref)--(ref) below, Definition (ref) admits a statistical implementation that is robust to estimation errors, jumps and microstructure noise.

remarkRoughness in the sense of Definition (ref) is a local property and is independent of the behavior of the underlying process as $T\to\infty$. In particular, if $(Z_t)_{t\geq0}$ is a stationary process, roughness is unrelated to the short- or long-memory properties of $Z$, which are usually defined in terms of the decay of its autocorrelation function $\rho_T = \mathbf{Cov}(Z_0,Z_T)$ as $T\to\infty$. As we show in Theorem (ref) below, roughness in fact is related to the autocorrelation of the increments of $Z$ as the sampling frequency increases but with the length of the time interval kept fixed. Of course, in parametric models, it can happen that roughness and long/short memory properties are parametrized by a single parameter. This is, for instance, the case with models based on fractional Brownian motion, where depending on whether the Hurst parameter $H$ is smaller or bigger than $\frac12$, one has either roughness and short memory or no roughness and long memory. The need to separate roughness on the one hand and short versus long memory on the other hand has been recognized in previous work already; see Bennedsen22 and LSY20.

AssumptionWe assume the following conditions under the null hypothesis: \begin{enumerate} • We have (ref)--(ref), where ${\alpha}$, ${\gamma}$, ${\Gamma}$, ${\alpha}^c$, ${\sigma}^c$, $\overline {\sigma}^c$, ${\gamma}^c$, ${\Gamma}^c$ and ${\alpha}^{\varphi}(u)$, ${\sigma}^{\varphi}(u)$, $\overline{\sigma}^{\varphi}(u)$, ${\gamma}^{\varphi}(u;\cdot)$ and ${\Gamma}^{\varphi}(u;\cdot)$ (for every $u\in\mathbb{R}$) are predictable and ${\alpha}^c$, ${\sigma}^c$ and $\overline{\sigma}^c$ are locally bounded, Moreover, $\inf\{c_s: 0\leq s\leq t\}>0$ for all $t>0$ almost surely. • There exist a sequence of stopping times $\tau_n$ increasing to infinity almost surely and nonnegative measurable functions $J_n(z)$ satisfying $\int_E J_n(z){\lambda}(dz)<\infty$ for all $n\in\mathbb{N}$ such that whenever $t\leq \tau_n$, we have \begin{equation} ( \lvert {\gamma}(t,z)\rvert^2+\lvert {\gamma}^c(t,z)\rvert^2+\lvert {\Gamma}(t,z)\rvert+\lvert {\Gamma}^c(t,z)\rvert)\wedge 1\leq J_n(z). \end{equation} • For any compact subset ${\cal U}\subseteq\mathbb{R}$, the process \begin{equation} t\mapsto \sup_{\delta\in(0,1)} \sup_{u\in{\cal U}} \bigl\{ \delta\lvert{\alpha}^{\varphi}(u/\sqrt{\delta})_t\rvert\} \end{equation} is locally bounded, while the processes \begin{align} & t\mapsto\sup_{s\in[0, \delta]} \mathbb{E}[\lvert {\alpha}_{t+s}-{\alpha}_t\rvert\wedge1\mid{\cal F}_t],\qquad t\mapsto\sup_{s\in[0, \delta]} \mathbb{E}[\lvert {\sigma}^c_{t+s}-{\sigma}^c_t\rvert\wedge1\mid{\cal F}_t],\\ & t\mapsto \sup_{u\in{\cal U}} \bigl\{ \lvert\delta {\sigma}^{\varphi}(u/\sqrt{\delta})_t\rvert+ \lvert\delta \overline{\sigma}^{\varphi}(u/\sqrt{\delta})_t\rvert \bigr\} \end{align} converge uniformly on compacts in probability to $0$ as $\delta\to0$. Finally, for any $n\in\mathbb{N}$, there are constants $C^n(\delta)\in(0,\infty)$ such that $C^n(\delta)\to0$ as $\delta\to0$ and \begin{align} \sup_{u\in{\cal U}} \bigl\{ \lvert \delta{\gamma}^{\varphi}(u/\sqrt{\delta}; t,z)\rvert^2 + \lvert\delta{\Gamma}^{\varphi}(u/\sqrt{\delta}; t,z)\rvert\bigr\}\wedge 1 \leq C^n(\delta) J_n(z) \end{align} for all $t\leq \tau_n$ and $z\in E$. \end{enumerate}

ExampleSuppose that $\int_0^t\int_E {\gamma}(s,z)(\mu-\nu)(ds,dz)=\int_0^t K_{s-} dL_s$, where $L$ is a mean-zero purely discontinuous martingale whose jump measure has $\mathbb{F}$-compensator ${\lambda}_{t-} dt F(dz)$, for some Lévy measure $F$ on $\mathbb{R}$. Further, suppose that $K$ and ${\lambda}$ are It\^o semimartingales. Then \begin{equation} {\varphi}(u)_t=\int_\mathbb{R} (e^{iu K_{t}z}-1-iuK_{t}z){\lambda}_t F(dz), \end{equation} and (ref), (ref) and (ref) are satisfied. The proof is given in Lemma (ref).

AssumptionWe have the following setup under the alternative hypothesis: \begin{enumerate} • We have (ref) and (ref), where $f$ and $g$ are as specified above and $\widetilde v$, ${\alpha}$, ${\gamma}$, ${\Gamma}$, ${\sigma}^v$ and $\overline {\sigma}^v$ are predictable processes. Moreover, ${\alpha}$, $\widetilde v$, ${\sigma}^v$ and $\overline{\sigma}^v$ are locally bounded and $\inf\{(f'(v_s))^2[({\sigma}^v_s)^2+(\overline {\sigma}^v_s)^2]: 0\leq s\leq t\}>0$ for all $t>0$ almost surely. • There exist stopping times $\tau_n$ increasing to infinity almost surely and nonnegative measurable functions $J_n(z)$ satisfying $\int_E J_n(z){\lambda}(dz)<\infty$ for all $n\in\mathbb{N}$ such that whenever $t\leq \tau_n$, we have \begin{equation} ( \lvert {\gamma}(t,z)\rvert^2+\lvert {\Gamma}(t,z)\rvert)\wedge 1\leq J_n(z). \end{equation} • For any compact subset ${\cal U}$ of $\mathbb{R}$, the process \begin{equation} t\mapsto \sup_{\delta\in(0,1)} \sup_{u\in{\cal U}} \bigl\{ \delta\lvert{\varphi}(u/\sqrt{\delta})_t\rvert\} \end{equation} is locally bounded. Moreover, for any $n\in\mathbb{N}$, there is $C_n\in(0,\infty)$ such that \begin{align} & \sup_{t\geq0} \mathbb{E}[\lvert\widetilde v_{t+\delta}-\widetilde v_t\rvert^2\mathbf 1_{\{t+\delta\leq \tau_n\}}]^{1/2} \leq C_n \delta^H h(\delta),\\ & \sup_{t\geq0} \sup_{u\in{\cal U}}\mathbb{E}\Bigl[ \lvert \delta{\varphi}(u/\sqrt{\delta})_{t+\delta}-\delta{\varphi}(u/\sqrt{\delta})_t\rvert^2\mathbf 1_{\{t+\delta\leq \tau_n\}} \Bigr]^{1/2} \leq C_n \delta^Hh(\delta),\\ &\sup_{t\geq0}\mathbb{E}[ (\lvert{\sigma}^v_{t+\delta}-{\sigma}^v_t\rvert^2+\lvert\overline{\sigma}^v_{t+\delta}-\overline{\sigma}^v_t\rvert^2)\mathbf 1_{\{t+\delta\leq \tau_n\}} ]^{1/2} \leq C_nh(\delta) \end{align} for all $\delta\in(0,1)$ and some $h$ satisfying $h(t)\to0$ as $t\to0$. \end{enumerate}

ExampleConsider the same setting as in Example (ref), except that $K$ and ${\lambda}$ can be any predictable locally bounded process satisfying \begin{equation} \sup_{s,t\geq0 } \mathbb{E}[\lvert K_t- K_s\rvert^2\mathbf 1_{\{s,t\leq \tau_n\}}]^{1/2}+\sup_{s,t\geq0 } \mathbb{E}[\lvert {\lambda}_t- {\lambda}_s\rvert^2\mathbf 1_{\{s,t\leq \tau_n\}}]^{1/2} \leq C_n \lvert t-s\rvert^H, \end{equation} for all $n\in\mathbb{N}$ and where $\tau_n$ and $C_n$ are as in Assumption (ref). Then, we still have (ref) and both (ref) and (ref) are satisfied. This is shown in Lemma (ref). In particular, this example covers the case where ${\lambda}_t= \ell(c_t)$ for some $C^1$-function $\ell$.

theoremA function $Z=(Z_t)_{t\in[0,T]}$ is rough in the sense of Definition (ref) if and only if \begin{equation} \rho(\Delta Z)^n_T=\frac{\sum_{i=1}^{n-1} \bigl(Z_{ \frac{i+1}n T}-Z_{\frac in T}\bigr)\bigl(Z_{\frac in T}-Z_{\frac {i-1}n T}\bigr)}{\sum_{i=1}^n \bigl(Z_{\frac in T}-Z_{\frac {i-1}n T}\bigr)^2} \stackrel{p}{\longrightarrow} \rho \end{equation} for some $ \rho<0$. In (ref), we let $0/0=1$.

AssumptionWe have \begin{equation} {u}^n_j = {\theta}/{\sqrt{\eta^n_j}},\qquad j=1,\dots,\lfloor T/(2p_n\Delta_n)\rfloor, \end{equation} where $\theta>0$ and $\eta^n_j$ is an ${\cal F}_{(2j-2)p_n\Delta_n}$-measurable random variable with the following properties: There is an adapted and stochastically continuous process $\eta_t$ such that $\inf\{ \eta_s:0\leq s\leq t\}>0$ almost surely for all $t>0$ and, for every $m\in\mathbb{N}$, \begin{equation} \sup_{j=1,\dots,\lfloor T/(2p_n\Delta_n)\rfloor}\mathbb{E}[\lvert\eta^n_j-\eta_{(2j-2)p_n\Delta_n}\rvert\mathbf 1_{\{(2j-2)p_n\Delta_n\leq \tau_m\}}] \to 0 \end{equation} as $n\to\infty$, where $(\tau_m)_{m\in\mathbb{N}}$ is the sequence of stopping times from Assumption (ref) or (ref). Moreover, for every $m\in\mathbb{N}$, there is are constants $0<\eta_m^-<\eta_m^+<\infty$ such that \begin{equation} \lim_{n\to\infty} \mathbb{P}\Bigl( \eta_m^-\leq \eta^n_j \leq \eta_m^+ for all j=1,\ldots,\lfloor T/(2p_n\Delta_n)\rfloor with (2j-2)p_n\Delta_n\leq \tau_m\Bigr) = 1. \end{equation}

ExampleAs we show in Lemma (ref), a natural choice is $\eta_t ={\sigma}^2_t$ together with \begin{equation} \eta^n_j= {\frac1{\ell_2-\ell_1+1}\sum_{\ell=\ell_1}^{\ell_2}\widehat{c}_{2j-\ell}^n},\qquad j=1,\ldots,\lfloor T/(2p_n\Delta_n)\rfloor, \end{equation} for some fixed $3\leq \ell_1\leq\ell_2$ and where \begin{equation} \widehat c^n_j=\frac{\pi}{2k_n\Delta_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \lvert\Delta^n_i x\Delta^n_{i-1} x\rvert \end{equation} is the bipower spot volatility estimator of barndorff2004power.

theoremSuppose that \begin{equation} {k_n\sqrt{\Delta_n}}\to0,\quad k_n\Delta_n^{1/2-\iota}\to\infty\quad for all \iota>0,\quad p_n/k_n \to \kappa \in[1,\infty), \end{equation} and consider the test statistic $\widehat T^n$ as defined in (ref), where $u^n_j$ satisfies Assumption (ref). \begin{enumerate} • Under the null hypothesis set forth in Assumption (ref), we have \begin{equation} \widehat{T}^n\stackrel{{\cal L}-s}{\longrightarrow}N(0,1). \end{equation} • Under the alternative hypothesis set forth in Assumption (ref), we have \begin{equation} \widehat{T}^n\stackrel{\mathbb{P}}{\longrightarrow}-\infty \quadat a rate of \sqrt{ 1/{(p_n\Delta_n)}}. \end{equation} \end{enumerate} In particular, a test with critical region \begin{equation} {\cal C}^n = \{\widehat T^n < \Phi^{-1}({\alpha})\}, \alpha\in(0,1), \end{equation} where $\Phi$ is the cumulative distribution function of the standard normal distribution, has asymptotic size ${\alpha}$ under Assumption (ref) and is consistent under Assumption (ref).

AssumptionThe observed log-prices are given by \begin{equation} y_{i\Delta_n}=x_{i\Delta_n} + {\epsilon}^n_i,\qquad i=0,\dots,\lfloor T/\Delta_n\rfloor, \end{equation} where the latent efficient price $x_{t}$ follows (ref) and satisfies Assumption (ref), while the noise variables ${\epsilon}^n_i$ are of the form \begin{equation} {\epsilon}^n_i = \Delta_n^\varpi \rho_{i\Delta_n} \chi_i. \end{equation} Here, $\varpi\in[0,\infty)$ determines the size of the noise, $\rho$ is an adapted locally bounded process and $(\chi_i)_{i\in\mathbb{Z}}$ is a strictly stationary $m$-dependent sequence of random variables (for some $m\geq0$), independent of ${\cal F}_\infty$ and with mean $0$, variance $1$ and finite moments of all orders. \begin{enumerate} • If $\varpi\leq \frac34$, we further assume that $\rho$ is an It\^o semimartingale of the form \begin{align} \rho_t &=\rho_0+\int_0^t{\alpha}^\rho_s ds + \int_0^t ({\sigma}^\rho_s dW_s + \widetilde{\sigma}^\rho_s d\widetilde W_s) + \int_0^t\int_E {\gamma}^\rho(s,z)(\mu-\nu)(ds,dz)\nonumber\\ &\quad + \int_0^t\int_E {\Gamma}^\rho(s,z)\mu(ds,dz), \end{align} where $\widetilde W$ is a standard $\mathbb{F}$-Brownian motion that is independent of $W$ (and jointly Gaussian and potentially correlated with $\overline W$, $W^{\varphi}$ and $\overline W^{\varphi}$), $\rho_0$ is an ${\cal F}_0$-measurable random variable, ${\alpha}^\rho$, ${\sigma}^\rho$ and $\widetilde{\sigma}^\rho$ are locally bounded adapted processes and ${\gamma}^\rho$ and ${\Gamma}^\rho$ are predictable functions. Moreover, as $\delta\to0$, the process $ t\mapsto \sup_{s\in[0,\delta]}\mathbb{E}[\lvert {\sigma}^\rho_{t+s}-{\sigma}^\rho_t\rvert\wedge 1 \mid {\cal F}_t]$ converges uniformly on compacts in probability to $0$ and we have $(\lvert{\gamma}^\rho(t,z)\rvert^2 + \lvert {\Gamma}^\rho(t,z)\rvert) \wedge1\leq J_n(z)$ whenever $t\leq \tau_n$ (where $\tau_n$ is the localizing sequence from Assumption (ref)). • If $\varpi\leq \frac12$, we have $\inf\{\rho_t: 0\leq s\leq t\}>0$ for all $t>0$ almost surely. If $\varpi\leq \frac12$ or if $\theta_0=0$ in Assumption (ref) below, there is $\varepsilon>0$ such that (ref) can be strengthened to \begin{equation} ( \lvert {\gamma}(t,z)\rvert^{2-\varepsilon}+\lvert {\gamma}^c(t,z)\rvert^{2-\varepsilon}+\lvert {\Gamma}(t,z)\rvert^{1-\varepsilon}+\lvert {\Gamma}^c(t,z)\rvert^{1-\varepsilon})\wedge 1\leq J_n(z) \end{equation} and the processes in (ref) and (ref) as well as the constant $C^n(\delta)$ from (ref) still converge to $0$ if divided by $\delta^\varepsilon$. \end{enumerate}

AssumptionWe have (ref), where $x_t$ satisfies Assumption (ref) and the noise variables ${\epsilon}^n_i$ satisfy (ref), where $\varpi$, $\rho$ and $(\chi_i)_{i\in\mathbb{Z}}$ have the properties listed in the sentence immediately after (ref). \begin{enumerate} • If $\varpi\leq\frac12+\frac{1}{2}H$, we further assume that \begin{equation} \rho_t=F(w_t),\quad w_t = w_0 + \int_0^t G(t-s)({\sigma}^{w}_sdW_s+\overline {\sigma}^w_sd\overline W_s+\widehat{\sigma}^{w}_s d\widehat W_s)+ \widetilde w_t, \end{equation} where $F$ is a $C^2$-function, $w_0$ is ${\cal F}_0$-measurable and $G(t)=K_{H_\rho}^{-1} t_+^{H_\rho-1/2}+G_0(t)$ for some $H_\rho\in(0,\frac12)$ and $G_0\in C^1([0,\infty))$ with $G_0(0)=0$. Moreover, $\widehat W$ is a standard $\mathbb{F}$-Brownian motion that is independent of $W$ and $\overline W$ and jointly Gaussian (and possibly correlated) with $W^{\varphi}$, $\overline W^{\varphi}$ and $\widetilde W$. The processes $\widetilde w$, ${\sigma}^w$, $\overline{\sigma}^w$ and $\widehat{\sigma}^w$ are adapted and locally bounded, and for any $n\in\mathbb{N}$ (and with $\tau_n$ as in (ref) and $C_n$ and $h$ as in (ref)), we have \begin{align} & \sup_{t\geq0 } \mathbb{E}[\lvert\widetilde w_{t+\delta}-\widetilde w_t\rvert^2\mathbf 1_{\{t+\delta\leq \tau_n\}} ]^{1/2} \leq C_n\delta^{H_\rho}h(\delta),\\ & \sup_{t\geq0}\mathbb{E}[ (\lvert{\sigma}^w_{t+\delta}-{\sigma}^w_t\rvert^2+\lvert\overline{\sigma}^w_{t+\delta}-\overline{\sigma}^w_t\rvert^2+\lvert\widehat{\sigma}^w_{t+\delta}-\widehat{\sigma}^w_t\rvert^2)\mathbf 1_{\{t+\delta\leq \tau_n\}} ]^{1/2} \leq C_nh(\delta). \end{align} • If $\varpi\leq \frac12$ or if $\theta_0=0$ in Assumption (ref) below, we have $h(t)=O(t^{\varepsilon})$ as $t\to0$ for some $\varepsilon>0$. • If $\varpi<\frac12+\frac12H$ and $(p_n\Delta_n)^H \Delta_n^{(1-2\varpi)_+} = o((p_n\Delta_n)^{H_\rho}\Delta_n^{(2\varpi-1)_+})$, then almost surely $\inf\{(\rho_sF'(w_s))^2[ ({\sigma}^w_s)^2+(\overline{\sigma}^w_s)^2+(\widehat{\sigma}^w_s)^2]: 0\leq s\leq t\}>0$ for all $t>0$. If $\varpi=\frac12$ and $H=H_\rho$, then almost surely $\inf\{z_s: 0\leq s\leq t\}>0$ for all $t>0$, where \begin{align*} z_t&=(\tfrac12 f'(v_t){\sigma}^v_t + \mathbb{E}[(\Delta\chi_1)^2]\rho_tF'(w_t){\sigma}^w_t)^2\\ &\quad+(\tfrac12 f'(v_t)\overline{\sigma}^v_t + \mathbb{E}[(\Delta\chi_1)^2]\rho_tF'(w_t)\overline{\sigma}^w_t)^2+( \mathbb{E}[(\Delta\chi_1)^2]\rho_tF'(w_t)\widehat{\sigma}^w_t)^2 \end{align*} and $\Delta\chi_i = \chi_i-\chi_{i-1}$. \end{enumerate}

AssumptionWe have Assumption (ref) with the following modifications: \begin{enumerate} • In (ref), $\theta=\theta_n$ is a sequence of positive numbers such that $\theta_n\to\theta_0\in[0,\infty)$. If $\varpi\leq \frac12$, we have $\theta_0=0$. The variables $\eta^n_j$ are ${\cal F}_{((2j-2)p_n-m-1)\Delta_n}$-measurable. • In (ref) and (ref), $\eta^n_j$ is replaced by $\eta^n_j/\Delta_n^{(2\varpi-1)\wedge 0}$, and (ref) still holds if the left-hand side is divided by $\Delta_n^\varepsilon$. The numbers $\varpi$ and ${\varepsilon}$ are the same numbers as in (ref) and (ref). • The process $ t\mapsto \sup_{s\in[0,\delta]}\mathbb{E}[\lvert \eta_{t+s}-\eta_t\rvert\wedge 1 ]/\delta^{\varepsilon}$ converges uniformly on compacts in probability to $0$ as $\delta\to0$. \end{enumerate}

theoremSuppose that Assumption (ref) is satisfied. If \begin{equation} \begin{split} &\frac{k_n\sqrt{\Delta_n}}{\theta_n^4}\to0, \qquad k_n\Delta_n^{1/2-\iota}\to\infty\quad for all \iota>0,\\ &p_n/k_n \to \kappa \in[1,\infty),\qquad k_n\leq p_n-m-1,\end{split} \end{equation} then the behavior of the test statistic $\widehat T^n$ remains the same as described in (ref) or (ref) depending on whether we have, respectively, Assumption (ref) or (ref). A test based on the critical region (ref) has asymptotic size ${\alpha}$ under Assumption (ref) and is consistent under (ref).

remarkWe conjecture that in Theorem (ref), the $m$-dependence assumption on $\chi$ can be weakened by requiring $\chi$ to be $\iota$-polynomially $\rho$-mixing for some $\iota>1$ as in jacod2017statistical, DX21 and li2022remedi.

remarkAs LPSY22 and SY23 show, there is a weak identification issue in estimating discrete-time rough volatility models. For example, within the class of ARFIMA models, it is asymptotically impossible to distinguish a near-stationary long-memory model from a rough model with near unit root dynamics. As our theoretical analysis shows, this weak identification issue is a feature of discrete-time volatility models only. In fact, roughness of a function is intrinsically a continuous-time concept and one can achieve identification of the roughness of the volatility path by considering an infill asymptotic setting. Therefore, our test can identify rough volatility irrespective of whether volatility has short or long memory behavior.

remarkAs we mentioned above, the hypotheses $H_0$ and $H_1$ in (ref) encompass most continuous-time volatility models considered in prior work. That said, there are specifications for $c_t$ that do not belong to either $H_0$ or $H_1$. The most notable such specification is one in which $c_t$ is a non-semimartingale process with finite quadratic variation. This is for instance the case if $c_t$ is a fractional process of the form (ref) but with $H>\frac12$ in (ref) (and $H_\rho>\frac12$ if there is noise). In this case, the volatility paths will have zero quadratic variation and the high-frequency increments of volatility will have positive autocorrelations. In fact, denoting $H_\ast= (H+(1-2\varpi)_+)\wedge (H_\rho+(2\varpi-1)_+)$, one can show that if $H_\ast\in(\frac12,\frac34)$, then $\widehat T^n\stackrel{\mathbb{P}}{\longrightarrow}+\infty$. This is because the positive autocovariance of volatility (or noise intensity) increments dominates in this case. If $H_\ast\in(\frac34,1)$, then estimation errors dominate and we have $\widehat T^n\stackrel{\mathcal{L}-s}{\longrightarrow} N(0,1)$. In summary, the asymptotic rejection rate of a one-sided test based on the critical region in (ref) will not exceed the nominal significance level under such very smooth specification for $c_t$. As we do not observe any significant positive values of the test statistic in the empirical application, we omit a formal proof of the aforementioned fact.

appendix\section{Proof of Main Results} \subsection{Proof of Theorems (ref) and (ref)} Since the noise-free case is obtained from the noisy one by choosing $\varpi=1$, $\rho=0$ and $m=-1$ (which we formally allow in the proof below), Theorem (ref) is just a special case of Theorem (ref). We sketch the main steps in the proof of the latter, with technical details deferred to Appendix (ref) in the supplement (supp). By a classical localization procedure (see e.g., JP12), it suffices to prove Theorem (ref) under the following Assumptions (ref), (ref) and (ref), which are strengthened versions of Assumptions (ref), (ref) and (ref), respectively. If $\theta_0=0$ in Assumption (ref), we write $a_n=o'(b_n)$ to denote $a_n/(b_n\Delta_n^\iota)\to0$ for some $\iota>0$ (that can vary from one place to another); if $\theta_0>0$, $a_n=o'(b_n)$ shall mean the same as $a_n=o(b_n)$. \let\oldtheAssumption#1 {SH$'_0$} \g@addto@macro\endAssumption{ \global\let#1\oldtheAssumption} \begin{Assumption} For any compact subset ${\cal U}$ of $\mathbb{R}$ and with the notation $0/0=0$, the following holds in addition to Assumption (ref) (the assumptions on ${\alpha}^\rho$, ${\sigma}^\rho$, $\widetilde{\sigma}^\rho$, ${\gamma}^\rho$ and ${\Gamma}^\rho$ only apply if $\varpi\leq \frac34$): \begin{enumerate} • We have ${\Gamma}^c\equiv {\Gamma}^\rho\equiv 0$ and the processes $x$, ${\alpha}$, $c$, ${\alpha}^c$, ${\sigma}^c$, $\overline{\sigma}^c$, $\rho$, ${\alpha}^\rho$, ${\sigma}^\rho$, $\widetilde {\sigma}^\rho$ and \begin{equation} t\mapsto\sup_{\delta\in(0,1)} \sup_{u\in{\cal U}} \bigl\{ \delta\lvert{\alpha}^{\varphi}(u/\sqrt{\delta})_t\rvert\} \end{equation} are uniformly bounded by a constant $K\in(0,\infty)$. Moreover, $c_t\geq K^{-1}$ and, if $\varpi\leq \frac12$, we have $\lvert\rho_t\rvert\geq K^{-1}$ for all $t>0$ almost surely. • There is a nonnegative measurable function $J(z)$ satisfying $J(z)\leq K$ as well as $\int_E J(z){\lambda}(dz)\leq K$ such that for all $z\in E$, \begin{equation} \sup_{{\omega}\in{\Omega}} \sup_{t\in[0,\infty)} \bigl\{ \lvert {\gamma}(t,z)\rvert^2+\lvert {\gamma}^c(t,z)\rvert^2+\lvert {\gamma}^\rho(t,z)\rvert^2 + \lvert {\Gamma}(t,z)\rvert\bigr\}\leq J(z). \end{equation} • As $\delta\to0$, the following sequences all converge to $0$: \begin{align} w(\delta)&=\begin{aligned}[t] &\sup_{{\omega}\in{\Omega},0\leq s\leq t,\lvert t-s\rvert \leq \delta} \bigl\{\mathbb{E}[\lvert {\alpha}_t-{\alpha}_s\rvert\mid{\cal F}_s]\\ &\qquad\qquad\qquad\qquad+ \mathbb{E}[\lvert {\sigma}^c_t-{\sigma}^c_s\rvert\mid{\cal F}_s]+ \mathbb{E}[\lvert {\sigma}^\rho_t-{\sigma}^\rho_s\rvert\mid{\cal F}_s] \bigr\},\end{aligned}\\ C_1(\delta)&=\sup_{{\omega}\in{\Omega}} \sup_{t\in[0,\infty)} \sup_{u\in{\cal U}} \bigl\{ \lvert \delta {\sigma}^{\varphi}(u/\sqrt{\delta})_t\rvert^2+\lvert \delta\overline{\sigma}^{\varphi}(u/\sqrt{\delta})_t\rvert^2 \bigr\}, \\ C_2(\delta)&=\sup_{{\omega}\in{\Omega}} \sup_{t\in[0,\infty)}\sup_{z\in E} \sup_{u\in{\cal U}} \bigl\{ \lvert \delta {\gamma}^{\varphi}(u/\sqrt{\delta}; t,z)\rvert^2/J(z) \bigr\},\\ C_3(\delta)&= \sup_{{\omega}\in{\Omega}}\sup_{t\in[0,\infty)} \sup_{z\in E} \sup_{u\in{\cal U}} \bigl\{ \lvert \delta{\Gamma}^{\varphi}(u/\sqrt{\delta};t,z)\rvert / J(z)\bigr\}. \end{align} • If $\varpi\leq\frac12$ or if $\theta_0=0$, we have \begin{equation} \sup_{{\omega}\in{\Omega}} \sup_{t\in[0,\infty)} \bigl\{ \lvert {\gamma}(t,z)\rvert^{2-\varepsilon}+\lvert {\gamma}^c(t,z)\rvert^{2-\varepsilon} + \lvert {\Gamma}(t,z)\rvert^{1-\varepsilon}\bigr\}\leq J(z),\qquad z\in E, \end{equation} and the constants in (ref)--(ref) still converge to $0$ after division by $\delta^\varepsilon$. \end{enumerate} \end{Assumption} \let\oldtheAssumption#1 {SH$'_1$} \g@addto@macro\endAssumption{ \global\let#1\oldtheAssumption} \begin{Assumption} The following holds in addition to Assumption (ref) (the assumptions on $\widetilde w$, ${\sigma}^w$, $\overline{\sigma}^w$, $\widehat{\sigma}^w$, $F$ and $G_0$ only apply if $\varpi\leq \frac12+\frac12H$): \begin{enumerate} • We have $g_0\equiv G_0\equiv0$. The functions $f$ and $F$ are bounded and twice differentiable with uniformly continuous and bounded derivatives and the processes $x$, ${\alpha}$, $v$, ${\sigma}^v$, $\overline{\sigma}^v$, $\widetilde v$, $w$, ${\sigma}^w$, $\overline {\sigma}^w$ and $\widehat {\sigma}^w$ and $\widetilde w$ are uniformly bounded by a constant $K\in(0,\infty)$. • We have $\lvert {\gamma}(t,z)\rvert^2 + \lvert {\Gamma}(t,z)\rvert\leq J(z)$, and if $\varpi\leq\frac12$ or if $\theta_0=0$, we have $\lvert {\gamma}(t,z)\rvert^{2-\varepsilon}+ \lvert {\Gamma}(t,z)\rvert^{1-\varepsilon}\leq J(z)$ for all $t\geq0$ and $z\in E$ (with ${\varepsilon}$ and $J$ as in Assumptions (ref) and (ref)). • For any compact subset ${\cal U}$ of $\mathbb{R}$, we have \begin{equation} \sup_{t\in(0,\infty)}\sup_{\delta\in(0,1)} \sup_{u\in{\cal U}} \bigl\{ \delta\lvert{\varphi}(u/\sqrt{\delta})_t\rvert\} <\infty. \end{equation} With the same $h$ as in Assumption (ref) and for all $t\geq0$ and $\delta>0$, we have \begin{align} & \mathbb{E}[\lvert\widetilde v_{t+\delta}-\widetilde v_t\rvert^2 ]^{1/2}\leq \delta^H h(\delta),\quad \mathbb{E}[\lvert\widetilde w_{t+\delta}-\widetilde w_t\rvert^2 ]^{1/2}\leq \delta^{H_\rho} h(\delta), \\ &\sup_{u\in{\cal U}} \mathbb{E} [\lvert \delta{\varphi}(u/\sqrt{\delta})_{t+\delta}-\delta{\varphi}(u/\sqrt{\delta})_t\rvert^2 ]^{1/2} \leq \delta^Hh(\delta),\\ \begin{split}&\mathbb{E}[\lvert{\sigma}^v_{t+\delta}-{\sigma}^v_t\rvert^2+\lvert\overline{\sigma}^v_{t+\delta}-\overline{\sigma}^v_t\rvert^2\\ &\qquad\qquad +\lvert{\sigma}^w_{t+\delta}-{\sigma}^w_t\rvert^2+\lvert\overline{\sigma}^w_{t+\delta}-\overline{\sigma}^w_t\rvert^2+\lvert\widehat{\sigma}^w_{t+\delta}-\widehat{\sigma}^w_t\rvert^2 ]^{1/2}\leq h(\delta).\end{split} \end{align} \end{enumerate} \end{Assumption} We can assume $g_0\equiv0$ because $\int_0^t g_0(t-s)( {\sigma}^v_sdW_s+\overline {\sigma}^vd\overline W_s)=\int_0^t\int_0^s g'_0(r-s)({\sigma}^v_sdW_s+\overline{\sigma}^v_sd\overline W_s)dr$ is just a drift that can be absorbed into $\widetilde v$. A similar argument shows that $G_0$ only contributes a drift that can be absorbed into $\widetilde w$, if $\varpi\leq \frac12+\frac12 H$. \let\oldtheAssumption#1 {SU$'$} \g@addto@macro\endAssumption{ \global\let#1\oldtheAssumption} \begin{Assumption} In addition to Assumption (ref), the following conditions are satisfied: \begin{enumerate} • There is $K\in(0,\infty)$ such that $\eta_t\geq K^{-1}$ for all $t\geq0$. • We have \begin{equation} \sup_{j=1,\dots,\lfloor T/(2p_n\Delta_n)\rfloor}\mathbb{E}[\lvert\eta^n_j/\Delta_n^{(2\varpi-1)\wedge0}-\eta_{(2j-2)p_n\Delta_n}\rvert] = o'(1) \end{equation} • We have \begin{equation} \sup_{t\geq0} \mathbb{E}[\lvert \eta_{t+\delta}-\eta_t\rvert] = o'(1). \end{equation} • There are constants $0<\eta_0^-<\eta_0^+<\infty$ such that \begin{equation} \lim_{n\to\infty} \mathbb{P}\Bigl(\eta_0^-\leq \eta^n_j /\Delta_n^{(2\varpi-1)\wedge0}\leq \eta_0^+ for all j=1,\ldots,\lfloor T/(2p_n\Delta_n)\rfloor\Bigr) = 1. \end{equation} \end{enumerate} \end{Assumption} We first consider the behavior of the test statistic under the null hypothesis and suppose that Assumptions (ref) and (ref) are satisfied. Using the notations \begin{align*} &\widetilde u^n_j=\frac{\theta_n}{ \sqrt{\widetilde\eta^n_j}}, \quad \widetilde \eta^n_j =\frac{\eta^n_j}{\Delta_n^{(2\varpi-1)\wedge0}},\quad \widetilde L_j^n(u) = \frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}e^{iu \Delta_i^ny/\Delta_n^{\varpi\wedge1/2}}, \\ &\widetilde c^n_j(u)=- \log \lvert\widetilde L^n_j(u)\rvert,\quad \widetilde {\frak c}^n_j(u)=\log \widetilde c^n_j(u),\quad \Delta^n_j \widetilde {\frak c}(u)=\widetilde {\frak c}^n_j(u)-\widetilde {\frak c}^n_{j-1}(u),\\ & \underline\Delta^n_j \widetilde {\frak c}(u,u')= \Delta^n_j \widetilde {\frak c}(u)- \Delta^n_{j-\mathds{1}} \widetilde {\frak c}(u'), \end{align*} we can write the test statistic $\widehat T^n$ as \begin{equation} \widehat{T}^n = \frac{\textstyle\sum_{j\in{\cal J}_n}\underline\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_j,\widetilde{u}^n_{j-\mathds{1}/2}) \underline\Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1},\widetilde u^n_{j-1-\mathds{1}/2})}{\textstyle\sqrt{\sum_{j\in{\cal J}_n}\bigl(\underline\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_j,\widetilde{u}^n_{j-\mathds{1}/2}) \underline\Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1},\widetilde u^n_{j-1-\mathds{1}/2})\bigr)^2}}=\frac{ V^n}{\sqrt{ W^n}}, \end{equation} where \begin{align} V^n&=\frac{ k_n }{\sqrt{\lvert{\cal J}_n\rvert }}\sum_{j\in{\cal J}_n}\underline\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_j,\widetilde{u}^n_{j-\mathds{1}/2}) \underline\Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1},\widetilde u^n_{j-1-\mathds{1}/2}),\\ W^n&= \frac{ k_n^2 }{\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n}\bigl(\underline\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_j,\widetilde{u}^n_{j-\mathds{1}/2}) \underline\Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1},\widetilde u^n_{j-1-\mathds{1}/2})\bigr)^2. \end{align} Since $\log \lvert z\rvert =\Re(\Log z)$, where $\Log$ is the principal branch of the complex logarithm, a first-order expansion gives \begin{equation} \Delta^n_{2j}\widetilde {\frak c}(\widetilde u^n_j)=\Re\bigl\{\Delta^n_{2j} \widetilde L(\widetilde u^n_j)/{\frak L}(\widetilde L^n_{2j-1}(\widetilde u^n_j))\bigr\} + higher-order terms, \end{equation} where $\Delta^n_i \widetilde L(u)= \widetilde L^n_j(u)-\widetilde L^n_{j-1}(u)$ and ${\frak L}(z)=z\log \lvert z\rvert$ for $z\in\mathbb{C}$. As $\widetilde L^n_j(u)$ is a local estimator of the conditional characteristic function of $\Delta^n_i y/\Delta_n^{\varpi\wedge 1/2}$, we can decompose \begin{equation} \widetilde{L}_j^n(u) = \underbrace{\widetilde{L}_j^{n,\mathrm{v}}(u)}_{variance} + \underbrace{\widetilde{L}_j^{n,\mathrm{b}}(u)}_{bias}+\underbrace{\widetilde{L}_j^{n,\mathrm{s}}(u)}_{signal}, \end{equation} where \begin{align*} \widetilde{L}_j^{n,\mathrm{v}}(u) &= \frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \bigl\{e^{i u_n \Delta_i^ny} -\mathbb{E} [e^{i u_n\Delta_i^ny} \mid{\cal F}_{(i-1)\Delta_n} ]\bigr \},\\ \widetilde{L}_j^{n,\mathrm{b}}(u) &=\frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}\mathbb{E} [e^{i u_n\Delta_i^ny}-e^{i u_n\Delta_{i,i-1}^ny}\mid{\cal F}_{(i-1)\Delta_n} ],\%=\frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}\Re\bigl\{\mathbb{E}[e^{iu_n \Delta_i^nx}\mid{\cal F}_{(i-1)\Delta_n} ]\bigr\} \widetilde{L}_j^{n,\mathrm{s}}(u) &=\frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}\mathbb{E} [e^{i u_n\Delta_{i,i-1}^ny}\mid{\cal F}_{(i-1)\Delta_n} ],\qquad u_n=u/\Delta_n^{\varpi\wedge1/2}, \end{align*} and $\Delta^n_{i,j} y=\Delta^n_{i,j} x+\Delta^n_{i,j} {\epsilon}$ with \begin{align*} \Delta^n_{i,j} x&= {\alpha}_{j\Delta_n}\Delta_n + {\sigma}_{j\Delta_n}\Delta^n_i W + \int_{(i-1)\Delta_n}^{i\Delta_n}\int_E {\gamma}(j\Delta_n,z) (\mu-\nu)(ds,dz)\\ & \quad+\int_{(i-1)\Delta_n}^{i\Delta_n} \int_E{\Gamma}(j\Delta_n,z) \mu(ds,dz) \end{align*} and $ \Delta^n_{i,j}{\epsilon}= \Delta_n^\varpi\rho_{j\Delta_n} \Delta \chi_i$. The next lemma is a key technical step in the proofs and shows that in the limit as $n\to\infty$, only the variance term in $\Delta^n_{2j}\widetilde L(\widetilde u^n_j)$ and the signal term in $\widetilde L^{n}_{2j-1}(\widetilde u^n_j)$ have to be retained in (ref) for the asymptotic analysis of $V^n$. In particular, the signal part of $\Delta^n_{2j}\widetilde L(\widetilde u^n_j)$ has no asymptotic contribution to $V^n$. To understand the intuition behind, consider the noise-free case where the signal part of $\Delta^n_{2j}\widetilde L(\widetilde u^n_j)$ is essentially given by an increment of $\exp(-\frac12(\widetilde u^n_j)^2c_t+\Delta_n {\varphi}(\widetilde u^n_j/\Delta_n)_t)$. Since this is an It\^o semimartingale in $t$ under the null hypothesis, it has asymptotically uncorrelated increments with variances dominated by $\Delta^n_{2j}\widetilde L^{\rm v}(\widetilde u^n_j)$ because of (ref). \begin{lemma} Under Assumptions (ref) and (ref), we have $ V^n -\widetilde V^n \stackrel{\mathbb{P}}{\longrightarrow} 0$ as $n\to\infty$, where \begin{equation} \begin{split} \widetilde V^n&=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Re\biggl\{\frac{ \Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1}))}-\frac{ \Delta^n_{2j-2-\mathds{1}} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1-\mathds{1}/2})}{{\frak L}({\cal L}^n_{j-1-\mathds{1}/2}(\widetilde u^n_{j-1-\mathds{1}/2}))}\biggr\}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\times\Re\biggl\{\frac{ \Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j})}{{\frak L}({\cal L}^n_j(\widetilde u^n_j))}-\frac{ \Delta^n_{2j-\mathds{1}} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-\mathds{1}/2})}{{\frak L}({\cal L}^n_{j-\mathds{1}/2}(\widetilde u^n_{j-\mathds{1}/2}))}\biggr\} \end{split} \end{equation} and ${\cal L}^n_j(u)=\exp(-\frac12u^2c_{((2j-2)p_n-m-1)\Delta_n}\Delta_n^{(1-2\varpi)_+})\Psi(u\Delta_n^{(\varpi-1/2)_+}\rho_{((2j-2)p_n-m-1)\Delta_n})$. \end{lemma} The asymptotic distribution of $\widetilde V^n$ can now be established by an application of the martingale central limit theorem. Recall that by assumption, $\theta_0=0$ if $\varpi\leq \frac12$, while for $\varpi>\frac12$, both $\theta_0=0$ and $\theta_0>0$ are allowed. \begin{lemma} Under Assumptions (ref) and (ref), we have \begin{equation} \widetilde V^n\stackrel{\mathcal{L}-s}{\longrightarrow} Q^{1/2} Z, \end{equation} where $Z$ is a standard normal random variable that is defined on a very good extension of the original probability space and independent of ${\cal F}_\infty$. The asymptotic ${\cal F}_\infty$-conditional variance $Q$ is given by \begin{equation} Q=\frac{1}{T} \int_{I_T} (q_t+q_{t-1})^2 dt , \end{equation} where $I_T=\bigcup_{k=1}^{T/2} [(2k-1),2k]$, \begin{equation} q_t=\begin{dcases} \frac{16\eta_t^2}{\theta_0^4c_t^2}\sinh^2(\tfrac12 \theta_0^2c_t/\eta_t) &if \theta_0>0,\\ \frac{4q_1(c_t\mathbf 1_{\{\varpi\geq\frac12\}},\rho_t\mathbf 1_{\{\varpi\leq \frac12\}})}{q_2(c_t\mathbf 1_{\{\varpi\geq\frac12\}},\rho_t\mathbf 1_{\{\varpi\leq \frac12\}})} &\text{if } \theta_0=0\end{dcases} \end{equation} and \begin{align*} q_1(c,\rho)&= c^2+2c\rho^2\mathbb{E}[(\Delta\chi_1)^2]+\rho^4\biggl(\frac12\mathbf{Var}((\Delta\chi_1)^2)+\sum_{r=1}^{m+1}\mathbf{Cov}((\Delta\chi_{r+1})^2,(\Delta\chi_1)^2)\biggr),\\ q_2(c,\rho)&= (c+\rho^2\mathbb{E}[(\Delta\chi_1)^2])^2. \end{align*} \end{lemma} Correspondingly, the denominator $W^n$ in (ref) is a consistent estimator of $Q$. \begin{lemma} Under Assumptions (ref) and (ref), we have $W^n\stackrel{\mathbb{P}}{\longrightarrow}Q$ as $n\to\infty$. \end{lemma} \begin{proof}[Proof of Theorem (ref) under (ref)] By localization, we can suppose that Assumptions (ref) and (ref) are in force. Since $Q$ is ${\cal F}_\infty$-measurable, (ref) immediately follows from Lemmas (ref)--(ref), the relation $\widehat T^n=V^n/\sqrt{W^n}$ and property (2.2.5) in JP12. \end{proof} For the proof of Theorem (ref) under the alternative hypothesis, we can suppose that Assumptions (ref) and (ref) hold true. A key difference between a semimartingale and a rough volatility process is that the latter has asymptotically correlated increments. So, instead of $V^n$ and $W^n$, the correctly normalized quantities are now \begin{equation} V^{n,\rm{alt}}=\frac{1}{\pi_n^2\lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n}\underline\Delta^n_{2j}\widetilde{{\frak c}} (u^n_{j},u^n_{j-\mathds{1}/2}) \underline\Delta^n_{2j-2}\widetilde{{\frak c}} (u^n_{j-1},u^n_{j-1-\mathds{1}/2}) \end{equation} and \begin{equation} W^{n,\rm{alt}}=\frac{1}{\pi_n^4\lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n}\bigl(\underline\Delta^n_{2j}\widetilde{{\frak c}} (u^n_{j},u^n_{j-\mathds{1}/2}) \underline\Delta^n_{2j-2}\widetilde{{\frak c}} (u^n_{j-1},u^n_{j-1-\mathds{1}/2})\bigr)^2, \end{equation} where $\pi_n=(p_n\Delta_n)^H\Delta_n^{(1-2\varpi)_+} + (p_n\Delta_n)^{H_\rho}\Delta_n^{(2\varpi-1)_+}$ and $H_\rho=0$ by convention if we have $\varpi>\frac12+\frac12 H$. \begin{lemma} Recall from (ref) that $\kappa$ is the asymptotic ratio of $p_n/k_n$. Furthermore, let ${\Lambda}=\lim_{n\to\infty} (p_n\Delta_n)^H\Delta_n^{(1-2\varpi)_+}/\pi_n$. (By our choice of $p_n$ in (ref), we either have ${\Lambda}=0$, ${\Lambda}=1$ or ${\Lambda}=\frac12$. The last case happens precisely when $H=H_\rho$ and $\varpi=\frac12$.) Under Assumptions (ref) and (ref), there are finite constants $C_{\kappa,H}$ and $\overline C_{\kappa,H}$ that only depend on $\kappa$ and $H$ such that \begin{equation}\begin{split} V^{n,\rm{alt}} &\stackrel{\mathbb{P}}{\longrightarrow}\frac{1}{T} (C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}})\int_{I_T} (A(t)+A(t-1)) dt, \\ W^{n,\rm{alt}} & \stackrel{\mathbb{P}}{\longrightarrow} \frac{2}{T}[1+2(C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}})^2] \int_{I_T}A(t)A(t-1)dt\\ &\qquad+\frac{1}{T} [\overline C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+\overline C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}}]\int_{I_T} ( B(t)+B(t-1)) dt, \end{split} \end{equation} where \begin{align*} A(t) &= \varsigma_t^{-4} \Bigl[( f'(v_t){\sigma}^v_t\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+2\mathbb{E}[(\Delta\chi_1)^2]\rho_t F'(w_t){\sigma}^w_t\mathbf 1_{\{{\Lambda}\in\{0,\frac12\}\}})^2\\ &\qquad\qquad +( f'(v_t)\overline{\sigma}^v_t\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+2\mathbb{E}[(\Delta\chi_1)^2]\rho_t F'(w_t)\overline{\sigma}^w_t\mathbf 1_{\{{\Lambda}\in\{0,\frac12\}\}})^2\\ &\qquad\qquad+(2\mathbb{E}[(\Delta\chi_1)^2]\rho_tF'(w_t)\widehat{\sigma}^w_t\mathbf 1_{\{{\Lambda}\in\{0,\frac12\}\}})^2\Bigr] \end{align*} and \begin{align*} B(t) &= \varsigma_t^{-4}\bigl[( f'(v_t){\sigma}^v_t\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+2\mathbb{E}[(\Delta\chi_1)^2]\rho_t F'(w_t){\sigma}^w_t\mathbf 1_{\{{\Lambda}\in\{0,\frac12\}\}})^2\\ & \qquad\qquad +( f'(v_t)\overline{\sigma}^v_t\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+2\mathbb{E}[(\Delta\chi_1)^2]\rho_t F'(w_t)\overline{\sigma}^w_t\mathbf 1_{\{{\Lambda}\in\{0,\frac12\}\}})^2\\ &\qquad\qquad+(2\mathbb{E}[(\Delta\chi_1)^2]\rho_tF'(w_t)\widehat{\sigma}^w_t\mathbf 1_{\{{\Lambda}\in\{0,\frac12\}\}})^2\bigr]^2 \end{align*} and $\varsigma_t^2=c_t\mathbf 1_{\{\varpi\geq\frac12\}}+\rho^2_t\mathbb{E}[(\Delta\chi_1)^2]\mathbf 1_{\{\varpi\leq \frac12\}}$. Moreover, $C_{\kappa,H}<0$ for all $\kappa\in[1,\infty)$ and $H\in(0,\frac12)$. \end{lemma} \begin{proof}[Proof of Theorem (ref) under (ref)] By localization, we may assume the stronger Assumptions (ref) and (ref). Since $\widehat T^n=\sqrt{\lvert{\cal J}_n\rvert} {V^{n,\rm{alt}}}/{\sqrt{W^{n,\rm{alt}}}}$, we obtain (ref) from Lemma (ref) and our assumptions (in particular, $A(t)$ is strictly negative and $B(t)$ is strictly positive). \end{proof} \subsection{Proofs for Examples (ref), (ref) and (ref)} \begin{lemma} In the setting of Example (ref), ${\varphi}(u)_t$ from (ref) satisfies (ref), (ref) and (ref). \end{lemma} \begin{proof} We only prove (ref); the other two properties can be shown similarly. We can assume that $K$ and ${\lambda}$ are driven by $W^{\varphi}$ and $\overline W^{\varphi}$ and by symmetry, it suffices to analyze ${\sigma}^{\varphi}$. Using It\^o's formula, we have that \[ {\sigma}^{\varphi}(u)_t =\int_\mathbb{R} \Bigl({\lambda}_t (e^{iuK_t z}-1)iuz{\sigma}^K_t+ (e^{iu K_{t}z}-1-iuK_{t}z){\sigma}^{\lambda}_t\Bigr)F(dz), \] where ${\sigma}^K$ and ${\sigma}^{\lambda}$ are the coefficients of $K$ and ${\lambda}$ with respect to $W$. Therefore \begin{equation}\begin{split} \sup_{t\in[0,T]}\sup_{u\in{\cal U}} \lvert\delta{\sigma}^{\varphi}(u/\sqrt{\delta})_t\rvert &\leq\int_\mathbb{R} \delta \sup_{t\in[0,T]}\sup_{u\in{\cal U}}\Bigl\lvert{\lambda}_t (e^{iuK_t z/\sqrt{\delta}}-1)iuz{\sigma}^K_t/\sqrt{\delta}\\ & \quad+ (e^{iu K_{t}z/\sqrt{\delta}}-1-iuK_{t}z/\sqrt{\delta}){\sigma}^{\lambda}_t\Bigr\rvert F(dz). \end{split}\end{equation} As $\delta\to0$, the integrand in the last line converges to $0$ pointwise in $z$ for all ${\omega}$. Moreover, it is bounded by \[ \biggl(\sup_{u\in{\cal U}} u^2\biggr)\biggl( \sup_{t\in[0,T]} \lvert{\lambda}_tK_t{\sigma}_t^K\rvert +\frac12 \sup_{t\in[0,T]}\lvert K_t^2{\sigma}^{\lambda}_t\rvert\biggr) z^2,\] which does not depend on $\delta$ and is integrable with respect to $F$ almost surely. Therefore, the dominated convergence theorem shows that (ref) tends to $0$ as $\delta\to0$. \end{proof} \begin{lemma} In the setup of Example (ref), the process ${\varphi}(u)_t$ from (ref) satisfies (ref) and (ref). \end{lemma} \begin{proof} Property (ref) follows immediately from (ref) and dominated convergence. For (ref), we can assume $\tau_1=\infty$ by localization and that $\lvert {\lambda}_t\rvert, \lvert K_t\rvert\leq C$. Then, because \begin{align*} \delta{\varphi}(u/\sqrt{\delta})_t-\delta{\varphi}(u/\sqrt{\delta})_s&=\int_\mathbb{R} \delta\Bigl(e^{iuK_sz/\sqrt{\delta}}(e^{iu(K_t-K_s)z/\sqrt{\delta}}- 1- iu(K_t-K_s)z/\sqrt{\delta})\\ &\quad\qquad\qquad+(e^{iuK_sz/\sqrt{\delta}}-1) iu(K_t-K_s)z/\sqrt{\delta}\Bigr){\lambda}_t F(dz)\\ &\quad+\int_\mathbb{R} \delta\Bigl( e^{iuK_sz/\sqrt{\delta}}- 1 - iuK_sz/\sqrt{\delta}\Bigr)({\lambda}_t-{\lambda}_s) F(dz), \end{align*} we have \begin{align*} &\delta^{-H}\sup_{s,t\geq0,\lvert t-s\rvert\leq \delta}\sup_{u\in{\cal U}} \mathbb{E}\Bigl[\bigl\lvert\delta{\varphi}(u/\sqrt{\delta})_t-\delta{\varphi}(u/\sqrt{\delta})_s\bigr\rvert^2\Bigr]\\ & \leq \int_\mathbb{R} \delta^{1-H}\sup_{s,t\geq0,\lvert t-s\rvert\leq \delta}\sup_{u\in{\cal U}}\mathbb{E}\Bigl[\Bigl\lvert\Bigl( e^{iuK_sz/\sqrt{\delta}}(e^{iu(K_t-K_s)z/\sqrt{\delta}}- 1- iu(K_t-K_s)z/\sqrt{\delta})\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad +(e^{iuK_sz/\sqrt{\delta}}-1) iu(K_t-K_s)z/\sqrt{\delta}\Bigr){\lambda}_t\\ & \quad+ ( e^{iuK_sz/\sqrt{\delta}}- 1 - iuK_sz/\sqrt{\delta} )({\lambda}_t-{\lambda}_s)\Bigr\rvert^2\Bigr]^{1/2} F(dz). \end{align*} Clearly, the integrand tends to $0$ pointwise in $z$ as $\delta\to0$. Since $\lvert e^{iux}-1\rvert \leq \lvert ux\rvert$, $\lvert e^{iux}-1-iux\rvert\leq \frac12 u^2 x^2$ and $\lvert K_t-K_s\rvert^2\leq 2C\lvert K_t-K_s\rvert$, we can further bound it by \begin{align*} &\delta^{-H}\sup_{s,t\geq0,\lvert t-s\rvert\leq \delta}\sup_{u\in{\cal U}}\mathbb{E}\biggl[\biggl(\frac12u^2\lvert K_t-K_s\rvert^2 \lvert {\lambda}_t\rvert z^2 + u^2\lvert K_s{\lambda}_t\rvert \lvert K_t-K_s\rvert z^2\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad \qquad\qquad+\frac12u^2K_s^2z^2\lvert {\lambda}_t-{\lambda}_s\rvert\biggr)^2\biggr]^{1/2} \\ &\quad\leq \delta^{-H}\sup_{s,t\geq0,\lvert t-s\rvert\leq \delta}\biggl(2 C^2\mathbb{E}[\lvert K_t-K_s\rvert^2]^{1/2} +\frac12 C^2\mathbb{E}[\lvert {\lambda}_t-{\lambda}_s\rvert^2]^{1/2} \biggr)\biggl(\sup_{u\in{\cal U}} u^2\biggr)z^2\\ &\quad\leq \frac52 C_1C^2\biggl(\sup_{u\in{\cal U}} u^2\biggr)z^2, \end{align*} which is $F$-integrable. The claim now follows by dominated convergence. \end{proof} \begin{lemma} If $\eta^n_j$ is chosen according to (ref) (with $\widehat c^n_j$ from (ref) computed using the observed prices $y$ in the noisy case), then Assumption (ref) is satisfied under both (ref) and (ref), while Assumption (ref) is satisfied under both (ref) and (ref). \end{lemma} \begin{proof} It suffices to show that Assumption (ref) is satisfied, as Assumption (ref) follows by considering the special case $\varpi=1$ and $\rho\equiv0$. To simplify notation, we further restrict ourselves to the case $\ell_1=\ell_2=2$ (i.e., $\eta^n_j=\widehat c^n_{2j-2}$). The general case is similar but more cumbersome to write. With this choice, $\eta^n_j$ is clearly ${\cal F}_{(2j-2)p_n\Delta_n}$-measurable. In order to show (ref) and (ref), we can work under the strengthened conditions of Assumption (ref) and (ref) by localization. In particular, we can assume $\tau_m=\infty$ for all $m\in\mathbb{N}$. Introducing the notation $\overline c^n_j = (\pi/2k_n\Delta_n)\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \lvert \Delta^n_i x^c+\Delta^n_i {\epsilon}\rvert \lvert \Delta^n_{i-1} x^c+\Delta^n_{i-1} {\epsilon}\rvert$, where $x^c=\int_0^t{\sigma}_sdW_s$, we have \begin{align*} \widehat c^n_j-\overline c^{n}_j &=\frac{\pi}{2k_n\Delta_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}\Bigl[( \lvert\Delta^n_i y\rvert-\lvert\Delta^n_i x^c+\Delta^n_i {\epsilon}\rvert)\lvert\Delta^n_{i-1} y\rvert\\ &\qquad\qquad\qquad\qquad\qquad + \lvert\Delta^n_i x^c+\Delta^n_i {\epsilon}\rvert(\lvert\Delta^n_{i-1} y\rvert -\lvert \Delta^n_{i-1}x^c+\Delta^n_{i-1} {\epsilon}\rvert)\Bigr]. \end{align*} Therefore, \begin{equation}\begin{split} &\mathbb{E}[\lvert \widehat c^n_j-\overline c^{n}_j\rvert/\Delta_n^{(2\varpi-1)\wedge0}]\\ &\quad\leq \frac{\pi}{2k_n\Delta_n^{2\varpi\wedge1}}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}\mathbb{E}\Bigl[\lvert\Delta^n_{i-1} y\rvert\mathbb{E}^n_{i-1}\bigl[\bigl\lvert \lvert\Delta^n_i y\rvert-\lvert\Delta^n_i x^c+\Delta^n_i {\epsilon}\rvert\bigr\rvert\bigr] \\ &\qquad+\mathbb{E}^n_{i-1}\bigl[ \lvert\Delta^n_i x^c+\Delta^n_i {\epsilon}\rvert\bigr]\bigl\lvert\lvert\Delta^n_{i-1} y\rvert -\lvert \Delta^n_{i-1}x^c+\Delta^n_{i-1} {\epsilon}\rvert\bigr\rvert\Bigr]. \end{split} \end{equation} As $\mathbb{E}^n_{i-1}\bigl[\bigl\lvert \lvert\Delta^n_i y\rvert-\lvert\Delta^n_i x^c+\Delta^n_i {\epsilon}\rvert\bigr\rvert\bigr]\leq \mathbb{E}[\lvert \Delta^n_i y-\Delta^n_i x^c-\Delta^n_i {\epsilon}\rvert]\leq (K+\int_E J(z){\lambda}(dz))\Delta_n+\sqrt{\Delta_n}h_3(\Delta_n)$ by Lemma (ref), we obtain $\mathbb{E}[\lvert \widehat c^n_j-\overline c^{n}_j\rvert/\Delta_n^{(2\varpi-1)\wedge0}]\leq Ch_3(\Delta_n)/\Delta_n^{(2\varpi-1)\wedge0}$, which goes to $0$ uniformly in $j$. With jumps and drift removed, it is now easy to show that we can replace $\Delta^n_{i-\ell} x^c$ by ${\sigma}_{(i-2)\Delta_n}\Delta^n_{i-\ell} W$ and $\Delta^n_{i-\ell}{\epsilon}$ by $\Delta_n^{\varpi}\rho_{((i-2)\Delta_n}\Delta^n_{i-\ell} \chi$ ($\ell=0,1$) in $\overline c^{n}_j/\Delta_n^{(2\varpi-1)\wedge0}$, incurring only an $O_p(\sqrt{\Delta_n})$- or $O_p(\Delta_n^H)$-error (depending on whether we have (ref) or (ref)) that is uniform in $j$. Using a martingale argument and writing $K(\chi)=\frac\pi2 \mathbb{E}[\lvert\Delta \chi_2\Delta\chi_1\rvert]$, one can prove that the resulting expression approaches, at a rate of $\sqrt{k_n}$, the term $k_n^{-1}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} ({\sigma}^2_{(i-2)\Delta_n}\Delta_n^{(1-2\varpi)\vee 0}+K(\chi)\rho^2_{(i-2)\Delta_n}\Delta_n^{(1-2\varpi)\wedge0})$, which approximates ${\sigma}^2_{(j-1)p_n\Delta_n+}\mathbf 1_{\{\varpi\geq\frac12\}}+K(\chi)\rho^2_{(j-1)p_n\Delta_n}\mathbf 1_{\{\varpi\leq \frac12\}}$ with a uniform $O_p(\sqrt{p_n\Delta_n})$- or $O_p((p_n\Delta_n)^H )$-error. This shows (ref) and its extension mentioned in Assumption (ref). Next, we consider (ref) and observe that $\widehat c^n_j = \widetilde c^{n,1}_j+\widetilde c^{n,2}_j$, where \begin{align*} \widetilde c^{n,1}_j &= \frac{\pi}{2k_n\Delta_n} \sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \mathbb{E}^n_{i-2}\bigl[\lvert \Delta^n_i y \Delta^n_{i-1} y\rvert \bigr], \\ \widetilde c^{n,2}_j &=\frac{\pi}{2k_n\Delta_n} \sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \Bigl\{\lvert \Delta^n_i y \Delta^n_{i-1} y\rvert - \mathbb{E}^n_{i-2}\bigl[\lvert \Delta^n_i y \Delta^n_{i-1} y\rvert \bigr]\Bigr\}. \end{align*} In $\widetilde c^{n,2}_j$, the $i$th term is ${\cal F}_{i\Delta_n}$-measurable and has a zero expectation conditionally on ${\cal F}_{(i-2)\Delta_n}$. So if we split the sum over $i$ into two, one taking only even and one only taken odd values of $i$, both will be martingale sums. Taking $p$th moments for $p\geq2$ and using the BDG inequality and ((ref)) or ((ref)), we obtain \begin{align*} \mathbb{E}[\lvert\widetilde c^{n,2}_j/\Delta_n^{(2\varpi-1)\wedge0}\rvert^p] &\leq \frac{C}{k_n^p\Delta_n^{(2\varpi \wedge 1)p}}\mathbb{E}\Biggl[\Biggl(\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \mathbb{E}^n_{i-2}[\lvert \Delta^n_i y \Delta^n_{i-1} y\rvert^2]\Biggr)^{p/2}\Biggr]\\ &=\frac{C}{k_n^p\Delta_n^{(2\varpi \wedge 1)p}}\mathbb{E}\Biggl[\Biggl(\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \mathbb{E}^n_{i-2}\Bigl[\mathbb{E}^n_{i-1}[\lvert \Delta^n_iy\rvert^2]\lvert \Delta^n_{i-1} y\rvert^2\Bigr]\Biggr)^{p/2}\Biggr] \\ &\leq \frac{C }{k_n^p\Delta_n^{(\varpi \wedge {1/2})p}} \mathbb{E}\Biggl[\Biggl(\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}\mathbb{E}^n_{i-2} [\lvert \Delta^n_{i-1} y\rvert^2]\Biggr)^{p/2}\Biggr] \leq \frac{C}{k_n^{p/2}}. \end{align*} Markov's inequality with $p=4$ implies that for any sequence $a_n\to0$, \begin{equation} \sum_{j=1}^{\lfloor T/(2p_n\Delta_n)\rfloor} \mathbb{P}(\lvert \eta^n_j-\widetilde c^{n,1}_{(2j-2)p_n\Delta_n} \rvert/\Delta_n^{(2\varpi-1)\wedge0} \geq a_n)\leq \frac{Cn}{a_n^4k_n^2p_n}. \end{equation} By (ref), the last line tends to $0$ if $a_n\to0$ slowly enough. In this case, it follows that \[ \lim_{n\to\infty} \mathbb{P}(\lvert \eta^n_j-\widetilde c^{n,1}_{(2j-2)p_n\Delta_n} \rvert /\Delta_n^{(2\varpi-1)\wedge0}\leq a_n \text{ for all } j=1,\dots,\lfloor T/(2p_n\Delta_n)\rfloor ) =1. \] This implies (ref) (with $\eta^n_j/\Delta_n^{(2\varpi-1)\wedge0}$ instead of $\eta^n_j$) provided that $\widetilde c^{n,1}_{(2j-2)p_n\Delta_n}/\Delta_n^{(2\varpi-1)\wedge0}$ is bounded from above and below, uniformly in ${\omega}$ and $j$. Thanks to ((ref)) or ((ref)), we have $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert^2]\leq C\Delta_n^{1\wedge 2\varpi}$, which readily gives the upper bound. For the lower bound, observe that for a general random variable $X$, we have $\mathbb{E}[\lvert X\rvert^{3/2}]=\mathbb{E}[\lvert X\rvert^{1/2} \lvert X\rvert]\leq \mathbb{E}[\lvert X\rvert]^{1/2}\mathbb{E}[\lvert X\rvert^2]^{1/2}$ by the Cauchy--Schwarz inequality, which implies \begin{equation} \mathbb{E}[\lvert X\rvert]\geq\frac{\mathbb{E}[\lvert X\rvert^{3/2}]^2}{\mathbb{E}[\lvert X\rvert^2]}. \end{equation} We want to apply this to $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert]$ in $\mathbb{E}^n_{i-2}\bigl[\lvert \Delta^n_i y \Delta^n_{i-1} y\rvert \bigr]=\mathbb{E}^n_{i-2}\bigl[\lvert \Delta^n_{i-1} y\rvert \mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert]\bigr]$. The denominator satisfies the bound $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert^2]\leq C\Delta_n^{1\wedge 2\varpi}$. For the numerator, we distinguish whether $\varpi\geq\frac12$ or $\varpi<\frac12$. In the first case, Jensen's inequality applied to the function $x\mapsto \lvert x\rvert^{3/2}$ for the conditional expectation $\mathbb{E}[\cdot\mid{\cal F}_\infty]$ yields $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert^{3/2}]=\mathbb{E}^n_{i-1}[\mathbb{E}[\lvert \Delta^n_i x+ \Delta^n_i{\epsilon}\rvert^{3/2}\mid{\cal F}_\infty]] \geq \mathbb{E}^n_{i-1}[\lvert \Delta^n_i x\rvert^{3/2}]$. Recalling ((ref)), we further have $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i x\rvert^{3/2}]^{2/3}\geq \mathbb{E}^n_{i-1}[\lvert \Delta^n_i x'\rvert^{3/2}]^{2/3}-\mathbb{E}^n_{i-1}[\lvert \Delta^n_i x''\rvert^{3/2}]^{2/3}\geq \mathbb{E}^n_{i-1}[\lvert\Delta^n_i x'\rvert^{3/2}]^{2/3}-(K+\int_E J(z){\lambda}(dz))\Delta_n$ by the reverse triangle inequality. Moreover, combining Doob's martingale inequality with the BDG inequality, we get \begin{align*} \mathbb{E}^n_{i-1}[\lvert \Delta^n_i x'\rvert^{3/2}] &\geq C\mathbb{E}^n_{i-1}\biggl[\sup_{s\in[(i-1)\Delta_n,i\Delta_n]}\lvert x'_s-x'_{(i-1)\Delta_n}\rvert^{3/2}\biggr]\\ &\geq C\mathbb{E}\Biggl[\Biggl(\int_{(i-1)\Delta_n}^{i\Delta_n} {\sigma}^2_s ds + \iint_{(i-1)\Delta_n}^{i\Delta_n} ({\gamma}(s,z)+{\Gamma}(s,z))^2\mu(ds,dz)\Biggr)^{3/4}\Biggr] \\ &\geq CK^{-3/4}\Delta_n^{3/4}. \end{align*} Therefore, if $n$ is sufficiently large, $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i x\rvert^{3/2}]\geq CK^{-3/4}\Delta_n^{3/4}$ and consequently, by (ref), $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert]\geq C\Delta_n^{1/2}$, where $C$ does not depend on $i$ or ${\omega}$. If $\varpi<\frac12$, we instead bound $\mathbb{E}^n_{i-1}[\lvert \Delta^n_i y\rvert]\geq \mathbb{E}^n_{i-1}[\lvert \Delta^n_i {\epsilon}\rvert]-\mathbb{E}^n_{i-1}[\lvert \Delta^n_i x\rvert]\geq \Delta_n^{\varpi}\rho_{(i-1)\Delta_n}\mathbb{E}[\lvert \Delta \chi_i\rvert] - C\Delta_n^{\varpi+1/2}-C\Delta_n^{1/2}\geq C\Delta_n^{\varpi}$. So in both cases, it follows that $\widetilde c^{n,1}_j /\Delta_n^{(2\varpi-1)\wedge0}\geq C$ where $C$ is independent of ${\omega}$ and $j$, which concludes the proof of (ref). \end{proof} \section{Proof of Auxiliary Results} Throughout this section, $C$ denotes a deterministic constant in $(0,\infty)$, whose value does not depend on any important parameters and may change from line to line. Given random variables $X_n$ and $Y_n$ and a deterministic sequence $a_n$, we write $X_n =Y_n +O(a_n)$ if $\lvert X_n-Y_n\rvert\leq Ca_n$, where $C$ neither depends on $n$ nor ${\omega}$. If $X_n$ and $Y_n$ further depend on indices such as $i$ or $j$, then $C$ does not depend on $i$ or $j$, either. The notations $X_n=Y_n+o(a_n)$ and $X_n=Y_n+o'(a_n)$ are used analogously. We also use the abbreviations $\int_a^b\int_E=\iint_a^b$ and $\mathbb{E}^n_i = \mathbb{E}[\cdot \mid {\cal F}_{i\Delta_n}]$. \subsection{Preliminary Estimates} \begin{lemma} Let $\phi(u)_t=\int_E e^{iu{\gamma}(s,z)}(e^{iu{\Gamma}(s,z)}-1){\lambda}(dz)$ and ${\cal U}\subseteq\mathbb{R}$ be a compact set. \begin{enumerate} • Under Assumption (ref), there is $C\in(0,\infty)$ such that for all $i$, \begin{equation} \begin{split} & \mathbb{E}^n_{i-1}[(\Delta^n_i x)^2] + \mathbb{E}^n_{i-1}[(\Delta^n_i c)^2]+ \mathbb{E}^n_{i-1}[(\Delta^n_i \rho)^2\mathbf 1_{\{\varpi\leq \frac34\}}]\leq C\Delta_n, \\ &\sup_{u\in{\cal U}}\Delta_n^2\mathbb{E}^n_{i-1}[(\Delta^n_i {\varphi}(u/\sqrt{\Delta_n}))^2]\leq C\Delta_n h_1(\Delta_n),\quad\mathbb{E}^n_{i-1}[(\Delta^n_i {\epsilon})^2] \leq C\Delta_n^{ 2\varpi}. \end{split}\end{equation} In the last line, $h_1$ is a function that satisfies $h_1(\Delta_n)=o'(1)$. • Under Assumption (ref), there is $C\in(0,\infty)$ such that for all $i$, \begin{equation} \begin{aligned} \mathbb{E}^n_{i-1}[(\Delta^n_i x)^2]& \leq C\Delta_n,& \mathbb{E}^n_{i-1}[(\Delta^n_i c)^2]&\leq C\Delta_n^{2H},\\ \mathbb{E}^n_{i-1}[(\Delta^n_i \rho)^2]&\leq C\Delta_n^{2H_\rho},& \mathbb{E}^n_{i-1}[(\Delta^n_i {\epsilon})^2]&\leq C\Delta_n^{ 2\varpi}. \end{aligned} \end{equation} • Under both Assumptions (ref) and (ref), \begin{equation} \begin{split} {\varphi}(n)&= \sup_{{\omega}\in{\Omega}}\sup_{t\in[0,\infty)}\sup_{u\in{\cal U}} \Delta_n\lvert{\varphi}(u/\sqrt{\Delta_n})_t\rvert =o'(1),\\ \phi(n)&= \sup_{{\omega}\in{\Omega}}\sup_{t\in[0,\infty)}\sup_{u\in{\cal U}} \Delta_n^{1/2}\lvert\phi(u/\sqrt{\Delta_n})_t\rvert =o'(1). \end{split} \end{equation} • Let $\Psi$ be the characteristic function of $\Delta\chi_1=\chi_1-\chi_0$. Then, for all $u,u'\in\mathbb{R}$, \begin{equation} \begin{split} \lvert\Psi(u)-1\rvert&\leq \tfrac12 u^2 \mathbb{E}[( \Delta\chi_1)^2],\\ \lvert \Psi(u)-\Psi(u')\rvert&\leq (\lvert u(u-u')\rvert+\tfrac12(u-u')^2)\mathbb{E}[(\Delta\chi_1)^2]. \end{split} \end{equation} \end{enumerate} \end{lemma} \begin{proof} Writing \begin{equation} x”_t = \int_0^t {\alpha}_s ds+\iint_0^t {\Gamma}(s,z){\lambda}(dz)ds,\qquad x'_t=x_t-x”_t, \end{equation} we have $\mathbb{E}^n_{i-1}[(\Delta^n_i x)^2]^{1/2}\leq \mathbb{E}^n_{i-1}[(\Delta^n_i x')^2]^{1/2}+\mathbb{E}^n_{i-1}[(\Delta^n_i x'')^2]^{1/2}$. Since $\lvert {\alpha}_t\rvert\leq K$ and $\lvert {\Gamma}(t,z)\rvert\leq J(z)$, we have $\mathbb{E}^n_{i-1}[(\Delta^n_i x'')^2]^{1/2}\leq (K+\int_E J(z){\lambda}(dz))\Delta_n\leq 2K\Delta_n$. Similarly, because ${\sigma}^2_t\leq C$ and $\lvert {\gamma}(t,z)\rvert^2\leq J(z)$, It\^o's isometry gives $\mathbb{E}^n_{i-1}[(\Delta^n_i x')^2]\leq (C+\int_E J(z){\lambda}(dz))\Delta_n\leq C\Delta_n$, showing the bound on $\Delta^n_i x$ in both (ref) and (ref). And because $\rho_t\leq K$, we have $\mathbb{E}^n_{i-1}[(\Delta^n_i {\epsilon})^2] \leq 4K^2\Delta_n^{2\varpi}$, which is the last inequality in both (ref) and (ref). Under Assumption (ref), the processes $c$ and $\rho$ (if $\varpi\leq\frac34$) are It\^o semimartingales with bounded coefficients, so the bounds on $\Delta^n_i c$ and $\Delta^n_i\rho$ in (ref) follow by applying the bound for $\Delta^n_i x$ to $x\in\{c,\rho\}$. A similar argument, combined with ((ref)) and ((ref))--((ref)), yields the bound on $\Delta^n_i {\varphi}(u/\sqrt{\Delta_n})$. Under Assumption (ref), the derivative $f'$ is bounded by some constant $C$, so the mean-value theorem implies that $\mathbb{E}^n_{i-1}[(\Delta^n_i c)^2] \leq C^2\mathbb{E}^n_{i-1}[(\Delta^n_i v)^2]\leq 2C^2\mathbb{E}^n_{i-1}[(\Delta^n_i \widetilde v)^2 + (\Delta^n_i \overline v)^2]$, where $\overline v_t=v_t-v_0-\widetilde v_t$. By ((ref)), we have $\mathbb{E}^n_{i-1}[(\Delta^n_i \widetilde v)^2]\leq \Delta_n^{2H}(h(\Delta_n))^2\leq C\Delta_n^{2H}$. Since ${\sigma}^v$ and $\overline{\sigma}^v$ are uniformly bounded by $K$ and $g_0\equiv0$, It\^o's isometry shows that \begin{equation}\begin{split} \mathbb{E}^n_{i-1}[(\Delta^n_i \overline v)^2] & = \int_0^{i\Delta_n} [g(i\Delta_n-s)-g((i-1)\Delta_n-s)]^2[({\sigma}^v_s)^2+(\overline{\sigma}^v_s)^2] ds\\ &\leq \frac{2K^2}{K_H^2}\int_0^{i\Delta_n} [(i\Delta_n-s)_+^{H-1/2}-((i-1)\Delta_n-s)_+^{H-1/2}]^2 ds\\ &= \frac{2K^2\Delta_n^{2H}}{K_H^2}\int_0^{i} [s^{H-1/2}-(s-1)_+^{H-1/2}]^2 ds\\ &\leq\frac{2K^2\Delta_n^{2H}}{K_H^2}\int_0^{\infty} [s^{H-1/2}-(s-1)_+^{H-1/2}]^2 ds. \end{split}\end{equation} The last integral is finite (in fact, equal to $K_H^2$ by Theorem 1.3.1 of Mishura08), which yields the bound on $\Delta^n_i c$ in (ref). The bound on $\Delta^n_i \rho$ in (ref) follows analogously. For the third part of the lemma, dominated convergence shows ${\varphi}(n)\to0$ because \begin{equation*} {\varphi}(n) \leq \int_E \sup_{{\omega}\in{\Omega}}\sup_{t\in[0,\infty)}\sup_{u\in{\cal U}} \Delta_n\lvert e^{iu{\gamma}(t,z)/\sqrt{\Delta_n}}-1-iu{\gamma}(t,z)/\sqrt{\Delta_n}\rvert {\lambda}(dz) \end{equation*} and the integrand tends to $0$ pointwise in $z$ and is bounded by $\frac12(\sup_{u\in{\cal U}} u^2) J(z)$, which is integrable with respect to ${\lambda}$ and independent of $n$. If $\theta_0=0$, we can use the estimate $\lvert e^{ix}-1-ix\rvert \leq C\lvert x\rvert^{2-{\varepsilon}}$ and ((ref)) to upgrade the previous bound to ${\varphi}(n)\leq C\Delta_n^{\varepsilon/2}\sup_{u\in{\cal U}}\lvert u\rvert^{2-{\varepsilon}} \int_E J(z) {\lambda}(dz) = o'(1)$. A similar argument shows $\phi(n)=o'(1)$. For the last part of the lemma, we deduce the first inequality from the bound $\lvert e^{iux}-1-iux\rvert\leq \frac12 (ux)^2$ and the assumption that $\mathbb{E}[\Delta\chi_1]=0$. To get the second inequality, we bound \begin{align*} \lvert \Psi(u')-\Psi(u)\rvert &= \lvert \mathbb{E}[e^{iu\Delta\chi_1}(e^{i(u'-u)\Delta\chi_1}-1)]\rvert\\ &\leq \lvert \mathbb{E}[ e^{iu\Delta\chi_1} (u'-u)\Delta\chi_1] \rvert + \tfrac12 (u'-u)^2\mathbb{E}[(\Delta\chi_1)^2]. \end{align*} Since $\lvert e^{iu\Delta\chi_1}-1\rvert\leq \lvert u\Delta\chi_1\rvert$ and $\Delta\chi_1$ is centered, we obtain the desired inequality. \end{proof} Next, we recall ((ref)) and introduce the notations ${\cal G}_i={\sigma}( \chi_j:j\leq i)$, ${\cal H}^n_i = {\cal F}_{i\Delta_n}\vee{\cal G}_i$ and $\overline\mathbb{E}^n_i[X]=\mathbb{E}[X\mid {\cal H}^n_i]$. The next lemma gathers estimates for the increments $\Delta^n_i \widetilde L^{\mathrm{v}}(u)=\widetilde L^{n,\rm v}_j(u)-\widetilde L^{n,\rm v}_{j-1}(u)$, $\Delta^n_i \widetilde L^{\mathrm{b}}(u)=\widetilde L^{n,\rm b}_j(u)-\widetilde L^{n,\rm b}_{j-1}(u)$ and $\Delta^n_i \widetilde L^{\mathrm{s}}(u)=\widetilde L^{n,\rm s}_j(u)-\widetilde L^{n,\rm s}_{j-1}(u)$. \begin{lemma} Grant Assumption (ref) and let ${\cal U}\subseteq\mathbb{R}$ be a compact set. There are constants $C\in(0,\infty)$ and, for any $p\geq2$, $C_p\in(0,\infty)$ such that the following holds for any $n\in\mathbb{N}$, $j=2,\dots,\lfloor T/(2p_n\Delta_n)\rfloor$, $\ell\in\{0,1\}$ and ${\cal H}^n_{((2j-2)p_n-m-1)}$-measurable random variable $U$ with values in ${\cal U}$: \begin{enumerate} • Let $[j]^m_n=jp_n-m-1$. Then \begin{equation} \overline \mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert \widetilde{L}_{2j-\ell}^{n,\mathrm{v}}(U)\rvert^p\bigr]\leq C_p/k_n^{p/2},\quad\overline \mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert \Delta^n_{2j}\widetilde{L}^{\mathrm{v}}(U)\rvert^p \bigr]\leq C_p/k_n^{p/2}. \end{equation} • For some function $h_2(t)$ satisfying $h_2(\Delta_n)=o'(1)$, we have \begin{equation} \overline\mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert \widetilde{L}_{2j-\ell}^{n,\mathrm{b}}(U)\rvert^p\bigr]\leq C_p \Delta_n^{p/2},\quad \overline\mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert \Delta^n_{2j} \widetilde{L}^{\mathrm{b}}(U)\rvert^p\bigr]\leq C_p (\Delta_n h_2(\Delta_n))^{p/2}. \end{equation} • Let $ \Theta(u)_t=iu\int_0^t {\alpha}_r dr- \frac12 u^2\int_0^t{\sigma}^2_r dr + \int_0^t{\varphi}(u)_r dr+\int_0^t \phi(u)_rdr$ and \begin{align*} \Delta^n_i \Theta(u)&=\Theta(u)_{i\Delta_n}-\Theta(u)_{(i-1)\Delta_n}, \\ \Delta\Theta(u)^n_{i}&=iu{\alpha}_{i\Delta_n}\Delta_n -\tfrac12 u^2 {\sigma}^2_{i\Delta_n}\Delta_n+\Delta_n{\varphi}(u)_{i\Delta_n} +\Delta_n\phi(u)_{i\Delta_n}. \end{align*} Then \begin{equation} \widetilde{L}_{2j-\ell}^{n,\mathrm{s}}(U) =\frac{1}{k_n}\sum_{i=(2j-\ell-1)p_n+1}^{(2j-\ell-1)p_n+k_n}e^{\Delta\Theta(U_n)^n_{i-1}} \Psi(U_n\Delta_n^{\varpi}\rho_{(i-1)\Delta_n}), \end{equation} and \begin{equation} \overline\mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert \Delta^n_{2j}\widetilde{L}^{\mathrm{s}}(U)\rvert^2 \bigr]=\mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert \Delta^n_{2j}\widetilde{L}^{\mathrm{s}}(U)\rvert^2 \bigr]\leq Cp_n\Delta_n. \end{equation} \end{enumerate} \end{lemma} \begin{proof} The identity (ref) is a simple consequence of the Lévy--Khintchine formula and the independence of $(\chi_i)_{i\in\mathbb{Z}}$ from ${\cal F}_\infty$. For all other statements, there is no loss of generality to assume that ${\cal U}=[0,u_\ast]$ for some $u_\ast\in(0,\infty)$. Let $Y^n(u)_t=e^{X^n(u)_t}$, $Y^{\prime n}(u)_t= e^{X^{\prime n}(u)_t}$, $\overline Y^n(u)_t=Y^{ n}(u)_tY^{\prime n}(u)_t$ and \begin{equation}\begin{split} X^n(u)_t&=-\tfrac12u^2\Delta_n^{(1-2\varpi)_+}{\sigma}^2_t+\Delta_n{\varphi}_t(u/\Delta_n^{\varpi\wedge 1/2})+\widetilde X^n(u)_t\mathbf 1_{\{\varpi\leq \frac34\}}, \\ X^{\prime n}(u)_t&=iu\Delta_n^{(1-\varpi)\vee 1/2}{\alpha}_t +\Delta_n\phi(u/\Delta_n^{\varpi\wedge 1/2})_t+\widetilde X^n(u)_t\mathbf 1_{\{\varpi> \frac34\}},\\ \widetilde X^n(u)_t&=\Log \Psi(u\Delta_n^{(\varpi-1/2)_+}\rho_t). \end{split}\end{equation} By (ref), we can decompose \begin{align} \Delta^n_{2j}\widetilde L^{\mathrm{s}}(U)&= \frac{1}{k_n}\sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} (\overline Y^n(U)_{(i-1)\Delta_n}-\overline Y^n(U)_{(i-p_n-1)\Delta_n})\nonumber \\ &=\frac{1}{k_n}\sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} \Bigl\{Y^{n}(U)_{(i-1)\Delta_n}( Y^{\prime n}(U)_{(i-1)\Delta_n}- Y^{\prime n}(U)_{(i-p_n-1)\Delta_n})\\ &\quad+ ( Y^{ n}(U)_{(i-1)\Delta_n}- Y^{ n}(U)_{(i-p_n-1)\Delta_n})Y^{\prime n}(U)_{(i-1)\Delta_n} \Bigr\}.\nonumber \end{align} Since $\rho$ takes the form ((ref)) if $\varpi\leq\frac34$, the mean-value theorem combined with (ref) yields \begin{equation} \mathbb{E}^n_{[2j-2]^m_n}[\lvert Y^n(U)_{(i-1)\Delta_n}-Y^n(U)_{(i-p_n-1)\Delta_n} \rvert^2]\leq C p_n\Delta_n. \end{equation} By assumption, ${\alpha}$ is continuous (if $\theta_0=0$, ${\varepsilon}$-H\"older continuous) in $L^2$, so by (ref) and (ref), \begin{equation} \mathbb{E}^n_{[2j-2]^m_n}[\lvert Y^{\prime n}(U)_{(i-1)\Delta_n}-Y^{\prime n}(U)_{(i-p_n-1)\Delta_n} \rvert^2]=o'(\Delta_n)=o'(p_n\Delta_n). \end{equation} Because $Y^n$ and $Y^{\prime n}$ are bounded by $1$, we obtain (ref). For (ref), note that $\xi^n_i = e^{iU_n\Delta^n_i y}-\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_i y}]$ is ${\cal H}^n_i$-measurable and because $(\Delta\chi_i)_{i\in\mathbb{Z}}$ is $(m+1)$-dependent and independent of ${\cal F}_\infty$, we have $\overline\mathbb{E}^n_{i-m-2}[e^{iU_n\Delta^n_i y}]=\mathbb{E}^n_{i-m-2}[e^{iU_n\Delta^n_i y}]$. This shows that $\overline\mathbb{E}^n_{i-m-2}[\xi^n_i]=0$. Upon writing \[ \widetilde{L}_{2j-\ell}^{n,\mathrm{v}}(U) =\sum_{i=1}^{m+2} \widetilde{L}_{2j-\ell,i}^{n,\mathrm{v}}(U),\quad\widetilde{L}_{2j-\ell,i}^{n,\mathrm{v}}(U)= \frac1{k_n}\sum_{k\geq0} \xi^n_{(2j-\ell-1)p_n+i+k(m+2)}\mathbf 1_{\{i+k(m+2)\leq k_n\}}, \] we realize that $\widetilde{L}_{2j-\ell,i}^{n,\mathrm{v}}(U)$ is a martingale sum for each $i$, relative to the discrete-time filtration $({\cal H}^n_{(2j-\ell-1)p_n+i+k(m+2)}: k\geq-1)$. Also, $\widetilde{L}_{2j-\ell,i}^{n,\mathrm{v}}(U)$ sums over at most $k_n/(m+2)$ nonzero terms and $\lvert \xi^n_i\rvert\leq2$. Therefore, combining the Minkowski and the Burkholder--Davis--Gundy (BDG) inequalities, we obtain the first inequality in (ref) from the estimates \begin{equation}\begin{split} &\mathbb{E}^n_{[2j-2]^m_n}[\lvert \widetilde{L}_{2j-\ell}^{n,\mathrm{v}}(U)\rvert^p]^{1/p}=\overline\mathbb{E}^n_{[2j-2]^m_n}[\lvert \widetilde{L}_{2j-\ell}^{n,\mathrm{v}}(U)\rvert^p]^{1/p}\leq \sum_{i=1}^{m+2}\overline\mathbb{E}^n_{[2j-2]^m_n}[\lvert \widetilde{L}_{2j-\ell,i}^{n,\mathrm{v}}(U)\rvert^p]^{1/p}\\ &\quad\leq C_pk_n^{-1}\sum_{i=1}^{m+2}\overline\mathbb{E}^n_{[2j-2]^m_n}\Biggl[\biggl(\sum_{k}\overline\mathbb{E}^n_{[2j-\ell-1]^m_n+i+k(m+2)-1}[\lvert\xi^n_{(2j-\ell-1)p_n+i+k(m+2)}\rvert^2 ] \biggr)^{p/2}\Biggr]^{1/p} \\ &\quad\leq 2 C_p\sqrt{(m+2)/k_n},\end{split} \end{equation} where $\sum_k$ sums over all $k$ such that $i+k(m+2)\leq k_n$ The second inequality in (ref) follows by the triangle inequality. Next, we decompose \begin{align} \mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_i y}-e^{iU_n\Delta^n_{i,i-1} y} ] &=\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}(e^{iU_n\Delta^n_i {\epsilon}}-e^{iU_n\Delta^n_{i,i-1} {\epsilon}})] \nonumber\\ &\quad+\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}{\epsilon}}(e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x})] \\ &\quad +\mathbb{E}^n_{i-1}[(e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x})(e^{iU_n\Delta^n_i {\epsilon}}-e^{iU_n\Delta^n_{i,i-1} {\epsilon}})].\nonumber \end{align} By Lemma (ref) and the Cauchy--Schwarz inequality, the last term is $o'(\sqrt{\Delta_n})$ and can therefore be neglected in the proof of (ref). Similarly, \begin{align*} &\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}(e^{iU_n\Delta^n_i {\epsilon}}-e^{iU_n\Delta^n_{i,i-1} {\epsilon}})]\\ &\quad=iU_n\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}e^{iU_n\Delta^n_{i,i-1} {\epsilon}}(\Delta^n_i {\epsilon}-\Delta^n_{i,i-1} {\epsilon})] + O(\Delta_n\mathbf 1_{\{\varpi\leq \frac34\}}+\Delta_n^{ (2\varpi-1)}\mathbf 1_{\{\varpi>\frac34\}}) \\ &\quad=iU \Delta_n^{(\varpi-1/2)_+}\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}e^{iU_n\Delta^n_{i,i-1} {\epsilon}}(\rho_{i\Delta_n}-\rho_{(i-1)\Delta_n})\chi_{i}]+o'(\sqrt{\Delta_n}). \end{align*} Since $(\chi_i)_{i\in\mathbb{Z}}$ is independent of ${\cal F}_\infty$, we obtain (using the notation $\widehat\Psi(u)=\mathbb{E}[e^{iu\Delta\chi_1}\chi_1]$) \begin{equation}\begin{split} &\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}(e^{iU_n\Delta^n_i {\epsilon}}-e^{iU_n\Delta^n_{i,i-1} {\epsilon}})]\\ &\quad=iU \Delta_n^{(\varpi-1/2)_+}\widehat\Psi(U\Delta_n^{(\varpi-1/2)_+}\rho_{(i-1)\Delta_n})\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}\Delta^n_i\rho]+o'(\sqrt{\Delta_n}), \end{split}\end{equation} where the $o'(\sqrt{\Delta_n})$-term is independent of ${\omega}$. If $\varpi> \frac34$, the mean-value theorem implies the bound $\lvert \Re\widehat\Psi(U\Delta_n^{(\varpi-1/2)_+}\rho_{(i-1)\Delta_n})-\Re\widehat\Psi(0)\rvert \leq \lvert \frac d{du} \Re \widehat\Psi(\widehat \rho_n) \rvert U\Delta_n^{\varpi-1/2}\lvert \rho_{(i-1)\Delta_n}\rvert\leq u_\ast K \lvert \mathbb{E}[\chi_1\Delta\chi_1]\rvert \Delta_n^{\varpi-1/2}$ for some intermediate value $\widehat\rho_n$ and an analogous bound for the imaginary part. Since $\widehat \Psi(0) = \mathbb{E}[\chi_1]=0$ and we have another factor $\Delta_n^{\varpi-1/2}$ in front of $\widehat\Psi$ in (ref), it follows that (ref) is $o'(\sqrt{\Delta_n})$ if $\varpi>\frac34$. If $\varpi \leq\frac34$, the last line in (ref) is $O(\sqrt{\Delta_n})$ because of (ref). Therefore, any additional modification that yields an extra $o'(1)$-term can be absorbed into the $o'(\sqrt{\Delta_n})$-bin in (ref). There are two cases: If $\frac12<\varpi\leq \frac34$, then the $\Delta_n^{\varpi-1/2}$-factor in front of $\widehat\Psi$ renders (ref) an $o'(\sqrt{\Delta_n})$-term. If $\varpi\leq \frac12$, we note that $\lvert {\alpha}^\rho\rvert\leq K$ and ${\Gamma}^\rho \equiv0$, so by Lemma (ref), $ \lvert\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}\Delta^n_i \rho]-\mathbb{E}^n_{i-1} [e^{iU_n\Delta^n_{i,i-1}x}\int_{(i-1)\Delta_n}^{i\Delta_n} ({\sigma}_s^\rho dW_s +\widetilde {\sigma}_s^\rho d\widetilde W_s) ] \rvert \leq K\Delta_n + \sqrt{\Delta_n}h_3(\Delta_n) $ almost surely, for some $h_3$ that satisfies $h_3(\Delta_n)=o'(1)$. In addition, because $\widetilde W$ is independent of $W$ and $w(\Delta_n),{\varphi}(n),\phi(n)=o'(1)$, we have \begin{align*} &\mathbb{E}^n_{i-1}\biggl[e^{iU_n\Delta^n_{i,i-1}x}\int_{(i-1)\Delta_n}^{i\Delta_n} ({\sigma}_s^\rho dW_s +\widetilde {\sigma}_s^\rho d\widetilde W_s)\biggr] =\mathbb{E}^n_{i-1}\biggl[e^{iU_n\Delta^n_{i,i-1}x}\int_{(i-1)\Delta_n}^{i\Delta_n} {\sigma}_s^\rho dW_s\biggr] \\ &\quad={\sigma}^\rho_{(i-1)\Delta_n}\mathbb{E}^n_{i-1} [e^{iU_n\Delta^n_{i,i-1}x}\Delta^n_i W]+O(\sqrt{\Delta_n}w(\Delta_n))\\ &\quad={\sigma}^\rho_{(i-1)\Delta_n}e^{iU_n{\alpha}_{(i-1)\Delta_n}\Delta_n+\Delta_n{\varphi}(U_n)+\Delta_n\phi(U_n)}\mathbb{E}^n_{i-1} [e^{iU_n{\sigma}_{(i-1)\Delta_n}\Delta^n_i W}\Delta^n_i W] + o'(\sqrt{\Delta_n})\\ &\quad=-i^{-1}e^{-\frac12 U^2\Delta_n^{1-2\varpi}{\sigma}^2_{(i-1)\Delta_n}}U{\sigma}_{(i-1)\Delta_n}{\sigma}^\rho_{(i-1)\Delta_n}\Delta_n^{1-\varpi}+ o'(\sqrt{\Delta_n}), \end{align*} where the last step follows from the fact that $U_n=U/\Delta_n^\varpi$ when $\varpi\leq\frac12$ and $\mathbb{E}[e^{iuX}X]=i^{-1}\frac{d}{du} \mathbb{E}[e^{iuX}]$. Since the last line above is $o'(\sqrt{\Delta_n})$ except when $\varpi=\frac12$, we have shown that \begin{equation}\begin{split} &\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}x}(e^{iU_n\Delta^n_i {\epsilon}}-e^{iU_n\Delta^n_{i,i-1} {\epsilon}})]\\ &\quad=-U^2e^{-\frac12 U^2{\sigma}^2_{(i-1)\Delta_n}} \widehat\Psi(U\rho_{(i-1)\Delta_n}){\sigma}_{(i-1)\Delta_n}{\sigma}^\rho_{(i-1)\Delta_n}\sqrt{\Delta_n}\mathbf 1_{\{\varpi=\frac12\}}+o'(\sqrt{\Delta_n}).\!\!\! \end{split}\end{equation} We pause here and move to the second term on the right-hand side of (ref), that is, \[\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}{\epsilon}}(e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x})]=\Psi(U_n\Delta_n^{\varpi}\rho_{(i-1)\Delta_n})\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x}].\] Writing ${\Lambda}(u)_t=iu\int_0^t {\sigma}_r dW_r+\int_0^t\int_E (e^{iu({\gamma}(r,z)+{\Gamma}(r,z))}-1)(\mu-\nu)(dr,dz)$ and using It\^o's formula (see JS03), one can verify that for any fixed $u$ and $s$, the process $Z(u,s)_{t}=e^{iu(x_t-x_s)-(\Theta(u)_t-\Theta(u)_s)}$ satisfies the stochastic differential equation $dZ(u,s)_t = Z(u,s)_{t-} d{\Lambda}(u)_t$ with $Z(u,s)_s=1$. In particular, since ${\Lambda}(u)$ is a martingale, so is $Z(u,s)$. Combining this with the Lévy--Khintchine formula and using the notation $Z^n_i(u)_t = Z(u,(i-1)\Delta_n)_t$, we obtain \begin{align}\nonumber \mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x} ] &=\mathbb{E}^n_{i-1}[e^{\Delta^n_i \Theta(U_n)}Z^n_i(U_n)_{i\Delta_n}] -e^{\Delta\Theta(U_n)^n_{i-1}} \\ & =\mathbb{E}^n_{i-1}[(e^{\Delta^n_i \Theta(U_n)}-e^{\Delta\Theta(U_n)^n_{i-1}})Z^n_i(U_n)_{i\Delta_n}] \\ &=e^{\Delta\Theta(U_n)^n_{i-1}}\mathbb{E}^n_{i-1}[(e^{\Delta^n_i \Theta(U_n)-\Delta\Theta(U_n)^n_{i-1}}-1)Z^n_i(U_n)_{i\Delta_n}]. \nonumber \end{align} As a consequence of our assumptions on ${\alpha}$, ${\sigma}^2$, ${\gamma}$ and ${\Gamma}$ and the elementary inequalities $\lvert U_n\rvert\leq u_\ast/\sqrt{\Delta_n}$, $\lvert e^{ix}-1\rvert \leq \lvert x\rvert$ and $\lvert e^{ix}-1-ix\rvert\leq \frac12x^2$, we have \begin{align*} & \lvert \Delta^n_i \Theta(U_n)\rvert\vee\lvert\Delta \Theta(U_n)^n_{i-1}\rvert\\ &\quad\leq u_\ast \sqrt{\Delta_n} K+\frac12 u_\ast^2K + \frac12u_\ast^2\int_E J(z){\lambda}(dz)+\lvert U\rvert\sqrt{\Delta_n} \int_E J(z){\lambda}(dz) \\ & \quad \leq 2K (u_\ast +\tfrac12 u_\ast^2 ). \end{align*} Denote the last term by $M$. Then $\lvert e^{\Delta \Theta(U_n)^n_{i-1}}\rvert\leq e^M$ and $\lvert Z^n_i(U_n)_{i\Delta_n}\rvert =e^{-\Re \Delta^n_i\Theta(U_n)} \leq e^M$, so \begin{equation}\begin{split} &\Bigl\lvert \mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x} ]-e^{\Delta\Theta(U_n)^n_{i-1}}\mathbb{E}^n_{i-1}[(\Delta^n_i \Theta(U_n)-\Delta\Theta(U_n)^n_{i-1}) Z^n_i(U_n)_{i\Delta_n}]\Bigr\rvert \\ &\quad\leq \tfrac12e^{2M}\mathbb{E}^n_{i-1}[\lvert \Delta^n_i \Theta(U_n)-\Delta\Theta(U_n)^n_{i-1}\rvert^2]. \end{split}\end{equation} Defining \begin{equation}\begin{split} \xi^{n,i}_1&=-\frac{U^2_n}2\int_{(i-1)\Delta_n}^{i\Delta_n} \int_{(i-1)\Delta_n}^{s} {\sigma}^c_rdW_rds,\\ \xi^{n,i}_2&=-\frac{U^2_n}2\int_{(i-1)\Delta_n}^{i\Delta_n}\int_{(i-1)\Delta_n}^s \overline{\sigma}^c_rd\overline W_rds,\end{split} \end{equation} and recalling $\phi(n)$ from (ref), we have \begin{equation}\begin{split} & \mathbb{E}^n_{i-1}[\lvert \Delta^n_i \Theta(U_n)-\Delta\Theta(U_n)^n_{i-1}-\xi^{n,i}_1-\xi^{n,i}_2\rvert] \\ &\quad\leq \frac{u_\ast}{\sqrt{\Delta_n}}\int_{(i-1)\Delta_n}^{i\Delta_n} \mathbb{E}^n_{i-1}[ \lvert {\alpha}_s-{\alpha}_{(i-1)\Delta_n}\rvert]ds +2\sqrt{\Delta_n}\phi(n)\\ & \quad +\Delta_n \sup_{s\in[(i-1)\Delta_n,i\Delta_n]} \mathbb{E}^n_{i-1}[\lvert {\varphi}(U_n)_s-{\varphi}(U_n)_{(i-1)\Delta_n}\rvert]+\frac12u_\ast^2 \int_{(i-1)\Delta_n}^{i\Delta_n} \mathbb{E}^n_{i-1}[\lvert {\alpha}^c_r\rvert] dr\\ & \quad +\frac12u_\ast^2 \mathbb{E}\biggl[\sup_{s\in[(i-1)\Delta_n,i\Delta_n]}\biggl\lvert \iint_{(i-1)\Delta_n}^s {\gamma}^c(r,z)(\mu-\nu)(dr,dz)\biggr\lvert\biggr]. \end{split}\end{equation} The first term on the right-hand side is bounded by $u_\ast \sqrt{\Delta_n}w(\Delta_n)$, where $w(\Delta_n)=o'(1)$. For the second term, note that $\phi(n)=o'(1)$ by (ref). The third term is bounded by \begin{equation}\begin{split} & \sup_{s\in[0,\infty)} \Biggl(\Delta_n^2\mathbb{E}^n_{i-1}[\lvert{\alpha}^{\varphi}(U_n)_s\rvert] + \Delta_n^{3/2}\mathbb{E}^n_{i-1}[{\sigma}^{\varphi}(U_n)_s^2+\overline{\sigma}^{\varphi}(U_n)_s^2]^{1/2}\\ &\qquad +\Delta_n^{3/2}\sup_{z\in E} \bigl(\mathbb{E}^n_{i-1}[{\gamma}^{\varphi}(U_n;s,z)^2]/J(z)\bigr)^{1/2}\biggl(\int_E J(z){\lambda}(dz)\biggr)^{1/2} \\ &\qquad + \Delta_n^2\sup_{z\in E} \lvert {\Gamma}^{\varphi}(U_n;s,z)/J(z)\rvert \int_E J(z){\lambda}(dz)\Biggr)\\ &\quad\leq K\Delta_n +\sqrt{C_1(\Delta_n)\Delta_n}+\sqrt{KC_2(\Delta_n)\Delta_n} + KC_3(\Delta_n)\Delta_n \end{split}\end{equation} due to ((ref)) and ((ref))--((ref)). Thanks to the bound $\lvert {\alpha}^c_r\rvert \leq K$ and Lemma (ref), the last two terms in (ref) are bounded by $C\Delta_n$ and $C\sqrt{\Delta_n}h_3(\Delta_n)$, respectively. Altogether, we have shown that the right-hand side of (ref) is $o'(\sqrt{\Delta_n})$. In a similar fashion, one can further show that the right-hand side of (ref) is $O({\Delta_n})$, which implies \begin{align*} &\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_{i,i-1}{\epsilon}}(e^{iU_n\Delta^n_i x}-e^{iU_n\Delta^n_{i,i-1} x})]\\ &\quad=\Psi(U_n\Delta_n^{\varpi}\rho_{(i-1)\Delta_n})e^{\Delta\Theta(U_n)^n_{i-1}}\mathbb{E}^n_{i-1}[(\xi^{n,i}_1+\xi^{n,i}_2)Z^n_i(U_n )_{i\Delta_n}] + o'(\sqrt{\Delta_n})\\ &\quad=\Psi(U_n\Delta_n^{\varpi}\rho_{(i-1)\Delta_n})e^{\Delta\Theta(U_n)^n_{i-1}}\mathbb{E}^n_{i-1}[\xi^{n,i}_1 Z^n_i(U_n )_{i\Delta_n}] + o'(\sqrt{\Delta_n}), \end{align*} almost surely. The last step holds because $Z^n_i(U_n )_t$ is a martingale driven by $W$ and $\mu-\nu$ (without any component driven by $\overline W$), so $Z^n_i(U_n )_t$ and $\xi^{n,i}_2$ are uncorrelated conditionally on ${\cal F}_{(i-1)\Delta_n}$. Since ${\sigma}^c$ satisfies ((ref)), we further have \begin{align*} &\mathbb{E}^n_{i-1}[\xi^{n,i}_1 Z^n_i(U_n )_{i\Delta_n}]\\ & =-\frac{U_n^2}2{\sigma}_{(i-1)\Delta_n}^c\mathbb{E}^n_{i-1}\biggl[ \int_{(i-1)\Delta_n}^{i\Delta_n} \int_{(i-1)\Delta_n}^s dW_sds Z^n_i(U_n )_{i\Delta_n}\biggr] +o'(\sqrt{\Delta_n})\\ & =-\frac{U_n^2}{2}{\sigma}^c_{(i-1)\Delta_n} \mathbb{E}^n_{i-1}\biggl[\int_{(i-1)\Delta_n}^{i\Delta_n} (i\Delta_n-r) dW_r\int_{(i-1)\Delta_n}^{i\Delta_n} Z^n_i(U_n )_{r-}d{\Lambda}(U_n)_r \biggr] +o'(\sqrt{\Delta_n}) \\ & =-\frac{iU_n^3}2{\sigma}^c_{(i-1)\Delta_n} \mathbb{E}^n_{i-1}\biggl[\int_{(i-1)\Delta_n}^{i\Delta_n} (i\Delta_n-r) dW_r\int_{(i-1)\Delta_n}^{i\Delta_n} Z^n_i(U_n )_r{\sigma}_rdW_r \biggr]+o'(\sqrt{\Delta_n})\\ & =-\frac{iU_n^3}2{\sigma}_{(i-1)\Delta_n}{\sigma}^c_{(i-1)\Delta_n} \int_{(i-1)\Delta_n}^{i\Delta_n} (i\Delta_n-r) \mathbb{E}^n_{i-1}[Z^n_i(U_n )_r]dr +o'(\sqrt{\Delta_n})\\ & =-\frac14iU^3\Delta_n^{(3/2-3\varpi)_+}{\sigma}_{(i-1)\Delta_n}{\sigma}^c_{(i-1)\Delta_n}\sqrt{\Delta_n}+o'(\sqrt{\Delta_n}). \end{align*} Recalling (ref) and (ref), we arrive at \begin{equation}\begin{split} &\mathbb{E}^n_{i-1}[e^{iU_n\Delta^n_i y}-e^{iU_n\Delta^n_{i,i-1}y}] \\ & \quad=-U^2e^{-\frac12 U^2{\sigma}_{(i-1)\Delta_n}^2} \widehat\Psi(U\rho_{(i-1)\Delta_n}){\sigma}_{(i-1)\Delta_n}{\sigma}^\rho_{(i-1)\Delta_n}\sqrt{\Delta_n}\mathbf 1_{\{\varpi=\frac12\}} \\ &\qquad-\frac{iU^3}4e^{-\frac12U^2{\sigma}^2_{(i-1)\Delta_n}} \Psi(U\Delta_n^{\varpi-1/2}\rho_{(i-1)\Delta_n}){\sigma}_{(i-1)\Delta_n}{\sigma}^c_{(i-1)\Delta_n}\sqrt{\Delta_n}\mathbf 1_{\{\varpi\geq\frac12\}}\\ &\qquad+o'(\sqrt{\Delta_n}), \end{split}\end{equation} where the $o'$-term does not depend on $i$ or ${\omega}$. This yields the first estimate in (ref). To prove the second estimate, denote the right-hand side of (ref) (without the $o'$-term) by $A^n_i$ and note that $\Delta^n_{2j}\widetilde L^{\rm b}(U)= \frac1{k_n}\sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} (A^n_i-A^n_{i-p_n})+o'(\sqrt{\Delta_n})$ by what we have shown so far. If $\varpi>\frac34$, (ref) implies $\lvert\Psi(U\Delta_n^{\varpi-1/2}\rho_{(i-1)\Delta_n})-\Psi (U\Delta_n^{\varpi-1/2}\rho_{(i-p_n-1)\Delta_n})\rvert\leq C \Delta_n^{2\varpi-1}=o'(\sqrt{\Delta_n})$. Because ${\sigma}$, ${\sigma}^c$ and, if $\varpi\leq \frac34$, $\rho$ and ${\sigma}^\rho$ are continuous (if $\theta_0=0$, at least ${\varepsilon}$-H\"older continuous) in $L^p$ and both $\Psi$ and $\widehat\Psi$ are differentiable (as $\chi$ has moments of all orders), it follows that $\mathbb{E}^n_{[2j-2]^m_n}[\lvert A^n_i-A^n_{i-p_n}\rvert^p]^{1/p}=o'(\sqrt{\Delta_n})$, proving the second estimate in (ref). \end{proof} \begin{lemma} Let $X_t=\iint_0^t {\gamma}^X(s,z)(\mu-\nu)(ds,dz)$, where ${\gamma}^X$ is a predictable function and satisfies $\lvert {\gamma}^X(s,z)\rvert^{2-{\varepsilon}\mathbf 1_{\{\theta_0=0\}}} \leq J(z)$ for all $s\geq0$, $z\in E$ and some measurable nonnegative function $J(z)$ with $\int_E J(z){\lambda}(dz)<\infty$. Then, for any $p\in[1,2)$, \[ \mathbb{E}\biggl[\sup_{t\in[s,s+\Delta_n]} \lvert X_t-X_s\rvert^p\biggr]^{1/p} \leq \sqrt{\Delta_n}h_3(\Delta_n), \] where $h_3$ does not depend on ${\gamma}^X$ or $s$ and satisfies $h_3(\Delta_n)=o'(1)$. \end{lemma} \begin{proof} Let $q=1$ if $\theta_0>0$ and $q=\frac{2}{2-\varepsilon}$ if $\theta_0=0$. There is no loss of generality to assume that $qp/2 < 1$. By the BDG inequality, \begin{align*} & \mathbb{E}\biggl[\sup_{t\in[s,s+\Delta_n]} \lvert X_t-X_s\rvert^p\biggr] \leq C\mathbb{E}\biggl[\biggl(\iint_{(i-1)\Delta_n}^{i\Delta_n} ({\gamma}^X(s,z))^2 \mu(ds,dz)\biggr)^{p/2}\biggr]\\ &\quad\leq C\Biggl\{\mathbb{E}\biggl[\biggl(\iint_{(i-1)\Delta_n}^{i\Delta_n} J(z)^q\mathbf 1_{\{J(z)\leq \Delta_n\}} \mu(ds,dz)\biggr)^{p/2}\biggr]\\ &\qquad\qquad\qquad\qquad\qquad+\mathbb{E}\biggl[\biggl(\iint_{(i-1)\Delta_n}^{i\Delta_n} J(z)^q\mathbf 1_{\{J(z)> \Delta_n\}} \mu(ds,dz)\biggr)^{p/2}\biggr]\Biggr\}\\ &\quad\leq C\Biggl\{\Delta_n^{p/2}\biggl(\int_E J(z)^q\mathbf 1_{\{J(z)\leq \Delta_n\}}{\lambda}(dz)\biggr)^{p/2}+\Delta_n\int_E J(z)^{qp/2}\mathbf 1_{\{J(z)>\Delta_n\}}{\lambda}(dz)\Biggr\}. \end{align*} For the last step, we applied Jensen's inequality to the first expectation and the bound $(a+b)^{p/2}\leq a^{p/2}+b^{p/2}$ to the second. Concerning the first term, note that $J(z)^q\leq \Delta_n^{q-1}J(z)$ if $J(z)\leq \Delta_n$. Moreover, since $J$ is integrable, $h_{31}(t)=\int_E J(z)\mathbf 1_{\{J(z)\leq t\}}{\lambda}(dz)$ satisfies $h_{31}(t)\to0$ as $t\to0$. Concerning the second term, we use H\"older's inequality to bound \begin{equation*} \Delta_n^{1-qp/2}\int_E J(z)^{qp/2}\mathbf 1_{\{J(z)>\Delta_n\}}{\lambda}(dz) \leq\biggl(\int_E J(z){\lambda}(dz)\biggr)^{qp/2} (\Delta_n{\lambda}(J(z)>\Delta_n))^{1-qp/2}. \end{equation*} Since $J$ is integrable and $t\mathbf 1_{\{J(z)>t\}}\leq t(J(z)/t)=J(z)$, the dominated convergence theorem implies that $h_{32}(t)=t{\lambda}(J(z)>t)$ satisfies $h_{32}(t)\to0$ as $t\to0$. The lemma is proved by choosing $h_3(t)=C^{1/p}t^{(q-1)/2}[\sqrt{h_{31}(t)}+C^{1/p}(\int_E J(z){\lambda}(dz))^{q/2}(h_{32}(t))^{1/p-q/2}]$. \end{proof} The moment estimates derived in the Lemma (ref) translate into pathwise bounds for the variance and bias terms $\widetilde L^{n,\mathrm{v}}_j(u)$ and $\widetilde L^{n,\mathrm{b}}_j(u)$ that hold with high probability. \begin{lemma} Under Assumptions (ref) and (ref), if we are given a compact set ${\cal U}$, we have $\mathbb{P}({\Omega}_n)\to1$, where ${\Omega}_n={\Omega}_n^{(1)}\cap{\Omega}_n^{(2)}$ and \begin{align*} {\Omega}_n^{(1)}&= \Bigl\{ \lvert \widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j)\rvert\vee \lvert \widetilde L^{n,\mathrm{b}}_{2j}(\widetilde u^n_j)\rvert \leq \Delta_n^{1/9} \text{ for all } \ell\in\{0,1\} \text{ and } j=1,\dots,\lfloor T/(2p_n\Delta_n)\rfloor\Bigr\},\\ {\Omega}_n^{(2)}&=\Bigl\{ \theta_n/\sqrt{\eta_0^+} \leq \widetilde u^n_j \leq \theta_n/\sqrt{\eta_0^-} \text{ for all } j=1,\dots,\lfloor T/(2p_n\Delta_n) \rfloor\Bigr\}. \end{align*} In particular, there are deterministic constants $C_1,C_2\in(0,\infty)$ such that for sufficiently large $n$, we have the following bounds on ${\Omega}_n$ for all $j=1,\dots,\lfloor T/(2p_n\Delta_n)\rfloor$: \begin{equation} \begin{split} C_1&\leq \lvert\widetilde L^n_{2j}(\widetilde u^n_j)\rvert,\lvert\widetilde L^n_{2j-1}(\widetilde u^n_j)\rvert,\lvert\widetilde L^{n,\mathrm{s}}_{2j}(\widetilde u^n_j)\rvert,\lvert\widetilde L^{n,\mathrm{s}}_{2j-1}(\widetilde u^n_j)\rvert\leq C_2,\\ C_1\theta_n^2&\leq \log\lvert\widetilde L^{n,\mathrm{s}}_{2j}(\widetilde u^n_j)\rvert^{-1}, \log\lvert\widetilde L^{n,\mathrm{s}}_{2j-1}(\widetilde u^n_j)\rvert^{-1} \leq C_2\theta_n^2. \end{split} \end{equation} \end{lemma} \begin{proof} Note that $\mathbb{P}({\Omega}_n^{(2)})\to1$ by ((ref)). On ${\Omega}_n^{(2)}$, we have $ \widetilde u^n_j \leq \overline \theta/\sqrt{\eta_0^-}$, where $\overline \theta = \sup_{n\in\mathbb{N}} \theta_n$. Therefore, by Lemma (ref) (note that $\widetilde u^n_j$ is ${\cal H}^n_{[2j-2]^m_n}$-measurable) and Markov's inequality, we have $ \sum_{j=1}^{\lfloor T/(2p_n\Delta_n)\rfloor}\mathbb{P} ( \lvert \widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j)\rvert \vee \lvert \widetilde L^{n,\mathrm{b}}_{2j-\ell}(\widetilde u^n_j)\rvert >\Delta_n^{1/9}\text{ for } \ell\in\{0,1\} ) \leq C\Delta_n^{-4/9}\lfloor T/(2p_n\Delta_n)\rfloor k_n^{-2}$, which implies $\mathbb{P}(({\Omega}_n^{(1)})^c\cap{\Omega}_n^{(2)})\to0$ by ((ref)). This yields the first statement of the lemma. For the first set of bounds in (ref), observe from (ref) that \begin{align*} \Re (\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1}) & \leq \tfrac12 (\widetilde u^n_j)^2 K + {\varphi}(n) +\phi(n)\leq C_2^2 K,\\ \lvert \Im(\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1})\rvert &\leq Ku^n_j\sqrt{\Delta_n}+{\varphi}(n) + \phi(n)=o(1) \end{align*} for large $n$. Hence, $ \Re e^{\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1}} = e^{\Re(\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1})} \cos\Im(\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2}) )$ and $\lvert \Im e^{\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1}}\rvert = e^{\Re(\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1})}\lvert\sin \Im(\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1})\rvert$ satisfy \begin{equation*} \tfrac12e^{-C_2^2K}\leq \Re e^{\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1}} \leq 1,\quad \lvert \Im e^{\Delta\Theta(\widetilde u^n_j/\Delta_n^{\varpi\wedge 1/2})^n_{i-1}}\rvert\leq KC_2\sqrt{\Delta_n}+{\varphi}(n)+\phi(n) \end{equation*} if $n$ is large. Moreover, $\lvert \widetilde u^n_j \Delta_n^{(\varpi-1/2)_+}\rho_{(i-1)\Delta_n}\rvert \leq K \Delta_n^{(\varpi-1/2)_+}\theta_n/\sqrt{\eta_0^-} \to 0$ on ${\Omega}_n^{(2)}$, which implies $\Psi( \widetilde u^n_j \Delta_n^{(\varpi-1/2)_+}\rho_{(i-1)\Delta_n})\to1$. By (ref), this gives $\frac14e^{-C_2^2K}\leq\lvert\widetilde L^{n,\mathrm{s}}_{2j}(\widetilde u^n_j)\rvert\leq 1$, which in turn shows $\frac18e^{-C_2^2K}\leq\lvert\widetilde L^{n}_{2j}(\widetilde u^n_j)\rvert\leq 1$ on ${\Omega}_n$. The bounds for $\lvert\widetilde L^n_{2j-1}(\widetilde u^n_j)\rvert$ and $\lvert\widetilde L^{n,\rm s}_{2j-1}(\widetilde u^n_j)\rvert$ can be derived in the same way. For the second set of inequalities in (ref), note that \begin{align*} \widetilde L^{n,\mathrm{s}}_{2j}(\widetilde u^n_j)&=k_n^{-1}\sum_{i=(2j-\ell-1)p_n+1}^{(2j-\ell-1)p_n+k_n}\overline Y^n(\widetilde u^n_j)_{(i-1)\Delta_n}\\ &=k_n^{-1}\sum_{i=(2j-\ell-1)p_n+1}^{(2j-\ell-1)p_n+k_n} Y^n(\widetilde u^n_j)_{(i-1)\Delta_n} + O(\sqrt{\Delta_n}) \end{align*} by (ref). As $ X^n(\widetilde u^n_j)_{(i-1)\Delta_n} = -\frac12 (\widetilde u^n_j)^2 c_{(i-1)\Delta_n}\mathbf 1_{\{\varpi\geq\frac12\}}+\Log \Psi(\widetilde u^n_j \rho_{(i-1)\Delta_n} )\mathbf 1_{\{\varpi\leq \frac12\}}+o(\theta_n^2)$, we can use (ref) to obtain $ \frac14(\theta_n^2/\eta_0^+)[ K^{-1}\mathbf 1_{\{\varpi\geq\frac12\}}+ K^{-2}\mathbb{E}[(\Delta\chi_1)^2]\mathbf 1_{\{\varpi\leq \frac12\}}]\leq \lvert X^n(\widetilde u^n_j)_{(i-1)\Delta_n}\rvert\leq (\theta_n^2/\eta_0^-)[ K^{-1}\mathbf 1_{\{\varpi\geq\frac12\}}+ K^{-2}\mathbb{E}[(\Delta\chi_1)^2]\mathbf 1_{\{\varpi\leq \frac12\}}]$, which completes the proof of (ref). \end{proof} \subsection{Proof of Technical Results from Appendix (ref)} \begin{proof}[Proof of Lemma (ref)] Expanding the product, we can decompose $V^n-\widetilde V^n$ into the sum of four differences, all of which can be treated similarly. Therefore, we only detail the proof that $V^{\prime n}-\widetilde V^{\prime n}\stackrel{\mathbb{P}}{\longrightarrow}0$, where \begin{equation}\begin{split} V^{\prime n}&=\frac{ k_n }{\sqrt{\lvert{\cal J}_n\rvert }}\sum_{j\in{\cal J}_n} \Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_j) \Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1}),\\ \widetilde V^{\prime n}&=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Re\biggl\{\frac{ \Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1}))}\biggr\}\Re\biggl\{\frac{ \Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j})}{{\frak L}({\cal L}^n_j(\widetilde u^n_j))}\biggr\}. \end{split}\end{equation} To this end, we further introduce \begin{equation} \overline V^{\prime n}=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Delta^n_{2j} \overline {\frak c}( \widetilde{u}^n_j) \Delta^n_{2j-2} \overline {\frak c}(\widetilde u^n_{j-1}), \end{equation} where $\Delta^n_j \overline {\frak c}(u)=\overline {\frak c}^n_j(u)-\overline {\frak c}^n_{j-1}(u)$ and \begin{equation}\begin{split} \overline {\frak c}^n_j(u)&=\log\log \lvert \widetilde L^{n,\mathrm{s}}_j(u)\rvert^{-1} + \Re \biggl\{ \frac{\widetilde L^{n,\mathrm{v}}_j(u)+\widetilde L^{n,\mathrm{b}}_j(u)}{ {\frak L}(\widetilde L^{n,\mathrm{s}}_j(u)) }\biggr\}\\ &\quad-\frac12\Re\biggl\{\frac{(\widetilde L^{n,\mathrm{v}}_j(u)+\widetilde L^{n,\mathrm{b}}_j(u))^2}{\widetilde L^{n,\mathrm{s}}_j(u){\frak L}(\widetilde L^{n,\mathrm{s}}_j(u))} \biggr\}-\frac12\biggl(\Re\biggl \{ \frac{\widetilde L^{n,\mathrm{v}}_j(u)+\widetilde L^{n,\mathrm{b}}_j(u)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_j(u)) }\biggr\}\biggr)^2. \end{split}\end{equation} Recalling the set ${\Omega}_n$ from Lemma (ref), we decompose $ V^{\prime n} =\overline V^{\prime n}+B^n_1+B^n_2 + B^n_3$, where \begin{align*} B^n_1&= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}(\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_j)-\Delta^n_{2j} \overline {\frak c}( \widetilde{u}^n_j)) \Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1})\mathbf 1_{{\Omega}_n}, \\ B^n_2 &=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Delta^n_{2j} \overline {\frak c}( \widetilde{u}^n_j) ( \Delta^n_{2j-2} \widetilde {\frak c}(\widetilde u^n_{j-1})- \Delta^n_{2j-2} \overline {\frak c}(\widetilde u^n_{j-1}))\mathbf 1_{{\Omega}_n} \end{align*} and $B^n_3=( V^{\prime n}-\overline V^{\prime n})\mathbf 1_{{\Omega}_n^c}$. By Lemma (ref), we have $ B^n_3 \to 0$ in probability. For the other two terms, we may assume that we are on the set ${\Omega}_n$. In fact, we will tacitly do so for the remainder of this section and write $\mathbb{E}[X\mathbf 1_{{\Omega}_n}]$ for $\mathbb{E}[X]$. Also, we will often use the fact that $o'(1)/\theta_n^4\to0$ by ((ref)). On ${\Omega}_n$, we have $\lvert{\frak L}( \widetilde L^{n,\rm s}_{2j-\ell}(\widetilde u^n_j))\rvert\geq C\theta_n^2$ and $\lvert \widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j)\rvert\vee\lvert \widetilde L^{n,\mathrm{b}}_{2j-\ell}(\widetilde u^n_j)\rvert\leq \Delta_n^{1/9}$ for all $\ell\in\{0,1\}$ and $j=1,\dots, \lfloor T/(2p_n\Delta_n)\rfloor$. So if $n$ is large, we have $\lvert (\widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j)+\widetilde L^{n,\mathrm{b}}_{2j-\ell}(\widetilde u^n_j))/{\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-\ell}(\widetilde u^n_j))\rvert \leq \frac12$. Upon realizing that $\overline{\frak c}^n_j(u)$ is a second-order expansion of $\widetilde {\frak c}^n_j(u)=\log\log \lvert \widetilde L^n_j(u)\rvert^{-1}=\log \Re \Log \widetilde L^n_j(u)^{-1}$ around $\widetilde L^{n,\rm s}_j(u)$, we can use Lemma (ref) to derive \begin{equation}\begin{split} \overline \mathbb{E}^n_{[2j-2]^m_n} \bigl[\lvert\widetilde {\frak c}^n_{2j-\ell}(\widetilde u^n_j)-\overline {\frak c}^n_{2j-\ell}(\widetilde u^n_j)\rvert^p\bigr]&\leq C\overline\mathbb{E}^n_{[2j-2]^m_n}\Biggl[\biggl\lvert\frac{\widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j)+\widetilde L^{n,\mathrm{b}}_{2j-\ell}(\widetilde u^n_j)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-\ell}(\widetilde u^n_j))}\biggr\rvert^{3p}\Biggr] \\ &\leq C(k_n^{-3p/2}+\Delta_n^{3p/2})/\theta_n^{6p}\leq Ck_n^{-3p/2}/\theta_n^{6p}. \end{split} \end{equation} Similarly, for large enough $n$ and because $p_n\Delta_n\leq 1/k_n$ by ((ref)), we have \[ \overline\mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert\Delta^n_{2j} \widetilde {\frak c}(\widetilde u^n_j)\rvert^2\bigr] \leq \overline\mathbb{E}^n_{[2j-2]^m_n}\biggl[ \biggl(\frac{\lvert\Delta^n_{2j} \widetilde L(\widetilde u^n_j)\rvert}{\textstyle\bigwedge_{\ell=0,1} \lvert {\frak L}( \widetilde L^n_{2j-\ell}(\widetilde u^n_j))\rvert }\biggr)^2\biggr]\leq Ck_n^{-1}/\theta_n^4. \] Analogously, $\overline\mathbb{E}^n_{[2j-2]^m_n}\bigl[ \lvert\Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)\rvert^2\bigr] \leq Ck_n^{-1}/\theta_n^4$. Hence, by the Cauchy--Schwarz inequality, we have $\mathbb{E} [ \lvert B^n_1\rvert ] \vee \mathbb{E} [\lvert B^n_2 \rvert ]\leq C\times k_n\sqrt{ p_n\Delta_n} \times 1/ (p_n\Delta_n) \times k_n^{-1/2}/\theta_n^2\times k_n^{-3/2}/\theta_n^6\leq C/\sqrt{k_n^3\Delta_n\theta_n^{16}}$, which converges to $0$ by ((ref)). Next, we want to replace $ \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)$ by a simpler expression $ \Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}$ ($\rm{s}$ for “simplified”), to be determined in several steps in the following, and accordingly replace $\overline V^{\prime n}$ by \begin{equation} \overline V^{\prime n}_{\rm s}= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )_{\rm s} \Delta^n_{2j-2}\overline{{\frak c}}( \widetilde{u}^n_{j-1} )_{\rm s}, \end{equation} Clearly, \begin{align*} \overline V^{\prime n}-\overline V^{\prime n}_{\rm s}&=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Bigl[( \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )- \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )_{\rm s}) \Delta^n_{2j-2}\overline{{\frak c}}( \widetilde{u}^n_{j-1} )\\ &\quad+ \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )_{\rm s} ( \Delta^n_{2j-2}\overline{{\frak c}}( \widetilde{u}^n_{j-1} )- \Delta^n_{2j-2}\overline{{\frak c}}( \widetilde{u}^n_{j-1} )_{\rm s})\Bigr], \end{align*} where $ \Delta^n_{2j-2}\overline{{\frak c}}( \widetilde{u}^n_{j-1} )$ is and $ \Delta^n_{2j-2}\overline{{\frak c}}( \widetilde{u}^n_{j-1} )_{\rm{s}}$ will be chosen as ${\cal H}^n_{(2j-3)p_n+k_n}$-measurable. Since $\overline\mathbb{E}^n_{[2j-2]^m_n}[ \lvert \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )\rvert]=O(k_n^{-1/2}/\theta_n^2)$ and $(2j-3)p_n+k_n\leq [2j-2]^m_n$ by ((ref)), we only have to make sure $\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}$ is chosen such that $\overline\mathbb{E}^n_{[2j-2]^m_n}[ \lvert \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )_{\rm s}\rvert]\leq Ck_n^{-1/2}/\theta_n^2$ on the one hand and $\overline\mathbb{E}^n_{[2j-2]^m_n}[\lvert \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )- \Delta^n_{2j}\overline{{\frak c}}( \widetilde{u}^n_j )_{\rm{s}}\rvert]=o(\sqrt{\Delta_n}\theta_n^2)$ on the other hand. Then the last display will converge to $0$ in $L^1$. To get the first version of $\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}$, we expand the increment $ \Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)$ using (ref) and omit all terms that are $o(\sqrt{\Delta_n}\theta_n^2)$. Simplifying terms, we arrive at $ \Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}= \Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)^{\rm I}_{\rm s}+\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)^{\rm II}_{\rm s}-\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)^{\rm III}_{\rm s}-\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)^{\rm IV}_{\rm s}-\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)^{\rm V}_{\rm s}$, where \begin{align*} \Delta^n_{j}\overline {\frak c}(u)_{\rm{s}}^{\rm I}&=\Re\Biggl\{\frac{\Delta^n_{j}\widetilde L^{\mathrm{s}}(u)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{j-1}(u))}\Biggr\}, \qquad\Delta^n_{j}\overline {\frak c}(u)_{\rm{s}}^{\rm II}= \Re\Biggl\{\frac{\Delta^n_{j} \widetilde L^{\mathrm{v}}(u)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{j-1}(u))}\Biggr\},\\ \Delta^n_{j}\overline {\frak c}(u)_{\rm{s}}^{\rm III}&= \Re\Biggl\{ \frac{(\widetilde L^{n,\rm v}_{j}(u))^2-(\widetilde L^{n,\rm v}_{j-1}(u))^2}{2\widetilde L^{n,\mathrm{s}}_{{j}-1}(u){\frak L}(\widetilde L^{n,\mathrm{s}}_{j-1}(u))}\Biggr\}, \qquad\Delta^n_{j}\overline {\frak c}(u)_{\rm{s}}^{\rm IV}=\Re\Biggl\{ \frac{\Delta^n_{j}\widetilde L^{\mathrm{s}}(u)\widetilde L^{n,\rm v}_{j}(u)}{({\frak L}(\widetilde L^{n,\mathrm{s}}_{j-1}(u)))^2}\Biggr\}, \\ \Delta^n_{j}\overline {\frak c}(u)_{\rm{s}}^{\rm V}&= \frac12\biggl[\biggl( \Re\biggl\{\frac{\widetilde L^{n,\mathrm{v}}_{j}(u)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{j-1}(u))}\biggr\}\biggr)^2-\biggl(\Re\biggl\{\frac{\widetilde L^{n,\mathrm{v}}_{j-1}(u)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{j-1}(u))}\biggr\}\biggr)^2\biggr] . \end{align*} We start by investigating the contribution of $\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm I}$. Our goal is to show that $ D^n_1=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Delta^n_{2j-2}\overline{\frak c} (\widetilde u^n_{j-1} )_{\rm s} \Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm I} $ is negligible. To this end, it suffices to show that \begin{equation*} \widetilde D^n_1=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Delta^n_{2j-2}\overline{\frak c} (\widetilde u^n_{j-1} )_{\rm s} \overline \mathbb{E}^n_{(2j-2)p_n}[\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm I}] \end{equation*} vanishes asymptotically. Indeed, since $ \Delta^n_{2j-2}\overline{\frak c} (\widetilde u^n_{j-1} )_{\rm s}$ is ${\cal H}^n_{(2j-2)p_n}$-measurable, the $j$th term that appears in the difference $D^n_1-\widetilde D^n_1$ will be ${\cal H}^n_{2jp_n}$-measurable with a zero ${\cal H}^n_{(2j-2)p_n}$-conditional expectation. Therefore, $D^n_1-\widetilde D^n_1$ is a martingale sum and consequently, we have $\mathbb{E}[\lvert D^n_1-\widetilde D^n_1\rvert^2]= O(k_n^2p_n\Delta_n\times 1/(p_n\Delta_n)\times k_n^{-1}/\theta_n^4\times p_n\Delta_n/\theta_n^4)=O(k_n^2\Delta_n/\theta_n^8)=o(1)$ by ((ref)). Next, consider the term $ \overline \mathbb{E}^n_{(2j-2)p_n}[\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm I}]= \mathbb{E}^n_{(2j-2)p_n}[\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm I}]$. Recalling (ref), we can use (ref) and (ref) to show that \begin{equation} 1-Y^{\prime n}(\widetilde u^n_j)_{(i-1)\Delta_n}=O(\sqrt{\Delta_n}). \end{equation} Moreover, $Y^n$ is $\frac12$-H\"older continuous in squared mean, so we have $\overline \mathbb{E}^n_{(2j-2)p_n}[\lvert\widetilde L^{n,\rm s}_{2j-1}(\widetilde u^n_j) - Y^n(\widetilde u^n_j)_{(2j-2)p_n\Delta_n}\rvert^2]\leq Cp_n\Delta_n$. Therefore, instead of $\widetilde D^n_1$, it is enough to consider \begin{align*} \widehat D^n_1&=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Delta^n_{2j-2}\overline{\frak c} (\widetilde u^n_{j-1} )_{\rm s} \Re\biggl\{\frac{ \mathbb{E}^n_{(2j-2)p_n} [\Delta^n_{2j}\widetilde L^{\mathrm{s}}(\widetilde u^n_j)]}{{\frak L}(Y^n(\widetilde u^n_j)_{(2j-2)p_n\Delta_n})}\biggr\}. \end{align*} Note that $X^n(u)$ from (ref) is an It\^o semimartingale with bounded coefficients (uniformly in sufficiently small $u$ such that $\Psi(uK)\geq\frac12$). Thus, by (ref), (ref) and a first-order expansion, \begin{align*} \mathbb{E}^n_{(2j-2)p_n} [\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)]& = \frac{Y^n(\widetilde u^n_j)_{(2j-2)p_n\Delta_n}}{k_n}\sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} \mathbb{E}^n_{(2j-2)p_n} [(X^n(\widetilde u^n_j)_{(i-1)\Delta_n}\\ &\quad -X^n(\widetilde u^n_j)_{(i-p_n-1)\Delta_n})] +O_p(p_n\Delta_n) = O_p(p_n\Delta_n), \end{align*} where $O_p(p_n\Delta_n)$ signifies a term whose ${\cal F}_{(2j-2)p_n\Delta_n}$-conditional $L^1$-norm is $O(p_n\Delta_n)=o'(\sqrt{\Delta_n})$, uniformly in ${\omega}$. This shows that $\widehat D^n_1$ is negligible. We proceed to $\Delta^n_{2j-2} \overline {\frak c}(\widetilde u^n_{j-1})_{\rm{s}}^{\rm I}$ and consider \begin{equation} \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Re\biggl\{ \frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{s}}(\widetilde u^n_{j-1})}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-3}(\widetilde u^n_{j-1}) )}\biggr\}\Delta^n_{2j}\overline{{\frak c}} (\widetilde u^n_j)_{\rm{s}}. \end{equation} Revisiting the proof of (ref), one can actually show $\mathbb{E}^n_{[2j-2]^m_n}[\lvert\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_{j})\rvert^2]\leq Cp_n\Delta_n\theta_n^2$.\footnote{This is because we can improve the right-hand side of (ref) to $Cp_n\Delta_n\theta_n^2$ if $U=\widetilde u^n_j$.} Since $\mathbb{E}[\lvert\Delta^n_{2j}\overline{{\frak c}} (\widetilde u^n_j)_{\rm{s}}^{\rm IV}\rvert^2]^{1/2}\leq C\sqrt{p_n\Delta_n/(k_n\theta_n^8)}\leq C\sqrt{\Delta_n}/\theta_n^4$, its contribution to (ref) is a term whose $L^1$-norm is $O( k_n\sqrt{p_n\Delta_n}\times 1/(p_n\Delta_n)\times\sqrt{p_n\Delta_n}\times \sqrt{ \Delta_n }/\theta_n^4)=O(\sqrt{k_n^2\Delta_n/\theta_n^8})$, which is negligible by ((ref)). Since $\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm I}$ can be omitted as we have shown, we can replace $\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}$ in (ref) by $\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm II}+\Delta^n_{2j}\overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm III}$. As above, because $Y^n(u)$ is an It\^o semimartingale with bounded coefficients (for $u$ sufficiently small) and $1-Y^{\prime n}(\widetilde u^n_j)_{(i-1)\Delta_n}=O(\sqrt{\Delta_n})$, we have $\mathbb{E}^n_{[2k-2]^m_n}[ \lvert\widetilde L^{n,\mathrm{s}}_{2k-\ell}(\widetilde u^n_k) - Y^n(\widetilde u^n_k)_{[2k-2]^m_n\Delta_n}\rvert^2]^{1/2}\leq C\sqrt{p_n\Delta_n}$ for $\ell\in\{0,1\}$ and $k\in\{j-1,j\}$. As a consequence, instead of (ref), it is sufficient to further analyze \begin{equation} \begin{split} &\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Re\biggl\{\frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{s}}(\widetilde u^n_{j-1})}{{\frak L}(Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})}\biggr\}\\ &\quad \times \Biggl(\Re\biggl\{\frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}}(\widetilde u^n_j)}{{\cal L}(Y^n(\widetilde u^n_j)_{[2j-2]^m_n\Delta_n})} +\frac{(\widetilde L^{n,\mathrm{v}}_{2j-1}(\widetilde u^n_j))^2-(\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j))^2}{2Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\\ &\quad +\frac12\biggl[\biggl(\Re\biggl\{\frac{\widetilde L^{n,\rm v}_{2j}(\widetilde u^n_j)}{{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr)^2-\biggl(\Re\biggl\{\frac{\widetilde L^{n,\rm v}_{2j-1}(\widetilde u^n_j)}{{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr)^2\biggr]\Biggr). \end{split} \end{equation} Since $\overline\mathbb{E}^n_{[2j-2]^m_n\Delta_n}[ \Delta^n_{2j}\widetilde L^{\mathrm{v}}(\widetilde u^n_j)]=0$, a martingale argument (along the lines of the analysis of $D^n_1$) reduces (ref) to \begin{equation}\begin{split} & \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Re\biggl\{\frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{s}}(\widetilde u^n_{j-1})}{{\frak L}(Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})}\biggr\} \Biggl(\Re\biggl\{ \frac{\overline\mathbb{E}^n_{[2j-2]^m_n}[ (\widetilde L^{n,\mathrm{v}}_{2j-1}(\widetilde u^n_j))^2-(\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j))^2]}{2Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\\ &\quad +\frac12\overline\mathbb{E}^n_{[2j-2]^m_n}\biggl[\biggl(\Re\biggl\{\frac{\widetilde L^{n,\rm v}_{2j}(\widetilde u^n_j)}{{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr)^2-\biggl(\Re\biggl\{\frac{\widetilde L^{n,\rm v}_{2j-1}(\widetilde u^n_j)}{{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr)^2\biggr]\Biggr).\end{split} \end{equation} Now, recall the notation $u_n=u/\Delta_n^{\varpi\wedge 1/2}$ and observe that for $\ell\in\{0,1\}$, \begin{equation}\begin{split} &\overline \mathbb{E}^n_{[2j-2]^m_n}[(\widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j))^2] =\frac1{k_n^2} \sum_{i=(2j-\ell-1)p_n+1}^{(2j-\ell-1)p_n+k_n} \overline\mathbb{E}^n_{[2j-2]^m_n}\Bigl[ \bigl( e^{i(\widetilde u^n_j)_n\Delta^n_i y } - \mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_i y}]\bigr)^2 \Bigr]\\ &\quad+\frac2{k_n^2} \sum_{i'<i} \overline\mathbb{E}^n_{[2j-2]^m_n}\Bigl[ \bigl( e^{i(\widetilde u^n_j)_n\Delta^n_i y } - \mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_i y}]\bigr) \bigl( e^{i(\widetilde u^n_j)_n\Delta^n_{i'} y } - \mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_{i'} y}]\bigr)\Bigr], \end{split} \end{equation} where the second sum is taken over all $(2j-\ell-1)p_n+1\leq i'<i\leq (2j-\ell-1)p_n+k_n$. The covariance term in (ref) is zero whenever $\lvert i-i'\rvert\geq m+2$. We want to replace $e^{i(\widetilde u^n_j)_n\Delta^n_i y} = e^{\Delta^n_i \Theta((\widetilde u^n_j)_n)}Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}e^{i(\widetilde u^n_j)_n\Delta^n_i {\epsilon}}$ everywhere in (ref) by $\widehat \xi^{n,i}_j=e^{\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n} \times e^{i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}}$, and similarly for the terms indexed by $i'$. As \begin{equation}\begin{split} \overline \mathbb{E}^n_{[2j-2]^m_n}\Bigl[\bigl\lvert ( e^{\Delta^n_i \Theta((\widetilde u^n_j)_n)}-e^{\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}})Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}e^{i(\widetilde u^n_j)_n\Delta^n_i {\epsilon}}\bigr\rvert^2\Bigr] &\leq C {p_n\Delta_n}, \\ \overline \mathbb{E}^n_{[2j-2]^m_n}\Bigl[\bigl\lvert e^{\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}(e^{i(\widetilde u^n_j)_n\Delta^n_i {\epsilon}}-e^{i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}})\bigr\rvert^2\Bigr] &\leq C {p_n\Delta_n} \end{split}\end{equation} with some $C$ that is uniform in ${\omega}$ and $j$,\footnote{If $\varpi\leq \frac34$, the second bound follows from the assumption that $\rho$ is an It\^o semimartingale with bounded coefficients. If $\varpi>\frac34$, we use the fact that $u_n\Delta_n^{\varpi} = \Delta_n^{\varpi-1/2}=o(\sqrt{p_n\Delta_n})$ by ((ref)).} we have \begin{equation}\begin{split} &\overline \mathbb{E}^n_{[2j-2]^m_n}[(\widetilde L^{n,\mathrm{v}}_{2j-\ell}(\widetilde u^n_j))^2] = \frac1{k_n^2} \sum_{i=(2j-\ell-1)p_n+1}^{(2j-\ell-1)p_n+k_n} \overline\mathbb{E}^n_{[2j-2]^m_n}\Bigl[ \bigl( \widehat \xi^{n,i}_j-\mathbb{E}^n_{i-1}[\widehat \xi^{n,i}_j]\bigr)^2\Bigr]\\ &\quad+\frac{2}{k_n^2} \sum_{i'<i}\overline \mathbb{E}^n_{[2j-2]^m_n}\Bigl[ \bigl( \widehat \xi^{n,i}_j-\mathbb{E}^n_{i-1}[\widehat \xi^{n,i}_j]\bigr)\bigl( \widehat \xi^{n,i'}_j-\mathbb{E}^n_{i-1}[\widehat \xi^{n,i'}_j]\bigr)\Bigr]+O(\sqrt{\Delta_n/k_n}). \end{split}\end{equation} Since $\mathbb{E}^n_{i-1}[\widehat\xi^{n,i}_j] = e^{\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}\Psi(\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+} \rho_{[2j-2]^m_n\Delta_n})$, we have \begin{equation}\begin{split} &\overline\mathbb{E}^n_{[2j-2]^m_n} [ ( \widehat \xi^{n,i}_j-\mathbb{E}^n_{i-1}[\widehat \xi^{n,i}_j] )^2 ] \\ & \quad =e^{2\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}\overline\mathbb{E}^n_{[2j-2]^m_n} [ e^{2i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}}( Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}-1)^2]\\ & \qquad+e^{2\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}\overline\mathbb{E}^n_{[2j-2]^m_n}[(e^{i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}}-\Psi(\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+} \rho_{[2j-2]^m_n\Delta_n}))^2]\\ & \qquad+e^{2\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}\overline\mathbb{E}^n_{[2j-2]^m_n}[e^{i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}}( Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}-1)\\ &\qquad \qquad\qquad\qquad\qquad\times(e^{i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}}-\Psi(\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+} \rho_{[2j-2]^m_n\Delta_n}))]. \end{split} \end{equation} Recall that $Z(u,s)_t=e^{iu(x_t-x_s)-(\Theta(u)_t-\Theta(u)_s)}$ is a martingale. By taking conditional expectation with respect to ${\cal F}_{[2j-2]^m_n\Delta_n}\vee {\cal G}_\infty$ and using the fact that $\overline\mathbb{E}^n_{[2j-2]^m_n}[Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}]= \mathbb{E}^n_{[2j-2]^m_n}[Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}]=1$, one can show that the third term is zero, while the first one on the right-hand side of (ref) is equal to $e^{2\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}\Psi(2\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+} \rho_{[2j-2]^m_n\Delta_n})$ times \begin{equation}\begin{split} & \mathbb{E}^n_{[2j-2]^m_n} [ ( Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}-1)^2]= \mathbb{E}^n_{[2j-2]^m_n} [( Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n})^2 +1 -2 Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}]\\ & \quad= \mathbb{E}^n_{[2j-2]^m_n} [e^{2i(\widetilde u^n_j)_n\Delta^n_i x -2\Delta^n_i\Theta((\widetilde u^n_j)_n)}]+1 -2 \mathbb{E}^n_{[2j-2]^m_n} [ Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}]\\ & \quad=e^{-2\Delta\Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}} \mathbb{E}^n_{[2j-2]^m_n} [e^{2i(\widetilde u^n_j)_n\Delta^n_i x}]-1 + O(\sqrt{p_n\Delta_n})\\ & \quad=e^{-2\Delta\Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}} \mathbb{E}^n_{[2j-2]^m_n} [e^{\Delta^n_i \Theta(2(\widetilde u^n_j)_n)}Z^n_i(2(\widetilde u^n_j)_n)_{i\Delta_n}]-1 + O(\sqrt{p_n\Delta_n})\\ & \quad=e^{\Delta\Theta(2(\widetilde u^n_j)_n)^n_{[2j-2]^m_n}-2\Delta\Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}-1+ O(\sqrt{p_n\Delta_n}), \end{split} \end{equation} where the $O$-term is uniform in ${\omega}$ and $j$. Note that the last line (apart from the $O(\sqrt{p_n\Delta_n})$-term) is independent of $i$ (and hence, of $\ell$), and so is the second term on the right-hand side of (ref) since $\chi$ is a stationary sequence. A similar argument shows that the same is true for the nondiagonal terms in (ref), which means that \begin{equation} \sup_{{\omega}\in{\Omega}}\sup_{j=2,\dots,\lfloor T/(2p_n\Delta_n)\rfloor}\Bigl\lvert\overline\mathbb{E}^n_{[2j-2]^m_n}\Bigl[(\widetilde L^{n,\mathrm{v}}_{2j-1}(\widetilde u^n_j))^2-(\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j))^2\Bigr]\Bigr\rvert= O(\sqrt{\Delta_n/k_n}). \end{equation} A similar argument proves that \begin{equation}\begin{split} &\sup_{{\omega}\in{\Omega}}\sup_{j=2,\dots,\lfloor T/(2p_n\Delta_n)\rfloor}\biggl\lvert\overline\mathbb{E}^n_{[2j-2]^m_n}\biggl[\biggl(\Re\biggl\{\frac{\widetilde L^{n,\rm v}_{2j}(\widetilde u^n_j)}{{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr)^2\\ & \qquad\qquad\qquad\qquad\qquad-\biggl(\Re\biggl\{\frac{\widetilde L^{n,\rm v}_{2j-1}(\widetilde u^n_j)}{{\frak L}(Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr)^2\biggr]\biggr\rvert = O(\sqrt{\Delta_n/k_n}/\theta_n^4). \end{split} \end{equation} Therefore, the $L^1$-norm of (ref) is $O(\sqrt{k_n\Delta_n/\theta_n^8})$, which shows that $\Delta^n_{2j-2} \overline {\frak c}(\widetilde u^n_{j-1})_s^{\rm I}$ has no asymptotic contribution. In other words, we have $\mathbb{E}[ \lvert \overline V^{\prime n} -\overline V^{\prime n}_{\rm s}\rvert]\to0$ where $\overline V_{\rm s}^n$ is given by (ref) and $ \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}= \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm II}- \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm III}- \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm IV}-\Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm V}$. Our next goal is to show that $\Delta^n_{2j}\overline{\frak c}(\widetilde u^n_j)^{\rm IV}_{\rm s}$ and $\Delta^n_{2j-2}\overline{\frak c}(\widetilde u^n_{j-1})^{\rm IV}_{\rm s}$ can also be omitted, that is, \begin{align*} &\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Re\Biggl\{ \frac{\widetilde L^{n,\mathrm{v}}_{2j-2}(\widetilde u^n_j)\Delta^n_{2j-2} \widetilde L^{\mathrm{s}}(\widetilde u^n_{j-1})}{({\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-3}(\widetilde u^n_j)))^2} \Biggr\}(\Delta^n_{2j} \overline{\frak c}(\widetilde u^n_j)^{\rm II}_{\rm s} +\Delta^n_{2j} \overline{\frak c}(\widetilde u^n_j)^{\rm III}_{\rm s} +\Delta^n_{2j} \overline{\frak c}(\widetilde u^n_j)^{\rm V}_{\rm s} )\\ &\quad+\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Delta^n_{2j-2}\overline {\frak c}(\widetilde u^n_{j-1})_{\rm s}\Re\Biggl\{ \frac{\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)}{({\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-1}(\widetilde u^n_j)))^2} \Biggr\} \end{align*} is asymptotically negligible. By Lemma (ref), one can readily see that $\Delta^n_{2j} \overline{\frak c}(\widetilde u^n_j)^{\rm III}_{\rm s} +\Delta^n_{2j} \overline{\frak c}(\widetilde u^n_j)^{\rm V}_{\rm s}$ do not contribute in the limit as $n\to\infty$. Furthermore, similarly to how we obtained (ref), we can replace $\widetilde L^{n,\mathrm{s}}_{2k-\ell}(\widetilde u^n_k)$ (for $\ell\in\{0,1\}$ and $k\in\{j-1,j\}$) by $Y^n(\widetilde u^n_k)_{[2k-2]^m_n\Delta_n}$. A martingale argument further allows us to take $\overline\mathbb{E}^n_{[2j-2]^m_n}[\cdot]$ inside the sum over $j$, which also eliminates $\Delta^n_{2j} \overline{\frak c}(\widetilde u^n_j)^{\rm II}_{\rm s}$. Hence, only \begin{align*} \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Delta^n_{2j-2}\overline{{\frak c}} (\widetilde u^n_{j-1})_{\rm{s}}\Re \Biggl\{ \frac{\overline\mathbb{E}^n_{[2j-2]^m_n}[\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)]}{({\frak L}(Y^n(\widetilde u^n_j)_{[2j-2]^m_n\Delta_n})^2} \Biggr\} \end{align*} has to be studied further. Here, we only need to keep the dominating part of $\Delta^n_{2j-2}\overline {\frak c}(\widetilde u^n_{j-1})_{\rm s}$ (all higher-order terms only have negligible contributions). Thus, we need to analyze \begin{equation} \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Re\Biggl\{ \frac{\Delta^n_{2j-2}\widetilde L^{\mathrm{v}}(\widetilde u^n_{j-1})}{{\frak L}(Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})}\Biggr\}\Re \Biggl\{ \frac{\overline\mathbb{E}^n_{[2j-2]^m_n}[\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)]}{({\frak L}(Y^n(\widetilde u^n_j)_{[2j-2]^m_n\Delta_n})^2} \Biggr\} . \end{equation} We decompose $ \Delta^n_j \widetilde L^{\mathrm{s}}(u)= \Delta^n_j \widetilde L^{\mathrm{s}}_1(u)+ \Delta^n_j \widetilde L^{\mathrm{s}}_2(u)+ \Delta^n_j \widetilde L^{\mathrm{s}}_3(u)$, where \begin{align*} \Delta^n_j \widetilde L^{\mathrm{s}}_1(u)&=\frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}e^{\Delta_n{\varphi}(u_n)_{(i-1)\Delta_n}}\Bigl\{e^{-\frac12 u^2_nc_{(i-1)\Delta_n}\Delta_n+(\Log \Psi(u_n\Delta_n^\varpi\rho_{(i-1)\Delta_n}))\mathbf 1_{\{\varpi\leq \frac34\}}}\\ &\qquad\qquad \qquad\qquad -e^{-\frac12 u^2_n c_{(i-p_n-1)\Delta_n}\Delta_n+(\Log \Psi(u_n\Delta_n^{\varpi} \rho_{(i-p_n-1)\Delta_n}))\mathbf 1_{\{\varpi\leq \frac34\}}}\Bigr\}, \\ \Delta^n_j \widetilde L^{\mathrm{s}}_2(u)&=\frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n}e^{-\frac12 u^2_n c_{(i-p_n-1)\Delta_n}\Delta_n+(\Log \Psi(u_n\Delta_n^\varpi \rho_{(i-p_n-1)\Delta_n}))\mathbf 1_{\{\varpi\leq \frac34\}}}\\ &\qquad\qquad\qquad \qquad\qquad\qquad\qquad\times(e^{\Delta_n{\varphi}(u_n)_{(i-1)\Delta_n}}-e^{\Delta_n{\varphi}(u_n)_{(i-p_n-1)\Delta_n}}),\\ \Delta^n_j \widetilde L^{\mathrm{s}}_3(u)&=\Delta^n_j \widetilde L^{\mathrm{s}}(u)-\Delta^n_j \widetilde L^{\mathrm{s}}_1(u)-\Delta^n_j \widetilde L^{\mathrm{s}}_2(u). \end{align*} Since $\Delta^n_j \widetilde L^{\rm s}_3(u)$ contains the contributions of drift, finite variation jumps and noise (if $\varpi>\frac34$), we have $\mathbb{E}^n_{(2j-2)p_n}[\lvert\Delta^n_{2j}\widetilde L^{\rm s}_3(\widetilde u^n_j)\rvert^2]^{1/2}=O(\sqrt{\Delta_n})$ by (ref). By (ref), we further have the bound $\mathbb{E}^n_{(2j-2)p_n}[\lvert\Delta^n_{2j}\widetilde L^{\rm s}_2(\widetilde u^n_j)\rvert^2]^{1/2}=o'(\sqrt{p_n\Delta_n})$. Therefore, in (ref), it suffices to keep $\Delta^n_{2j}\widetilde L^{\rm s}_1(\widetilde u^n_j)$ instead of $\Delta^n_{2j}\widetilde L^{\rm s}(\widetilde u^n_j)$. Writing ${\cal L}^n(u;c,\rho)=e^{-\frac12 u^2_n \Delta_n c+(\Log \Psi(u_n\Delta_n^\varpi\rho))\mathbf 1_{\{\varpi\leq 3/4\}}}$, we have that $\Delta^n_{2j}\widetilde L^{\mathrm{s}}_1(\widetilde u^n_j)$---up to an error whose $\mathbb{E}^n_{(2j-2)p_n}[(\cdot)^2]^{1/2}$-norm is $o'(\sqrt{p_n\Delta_n})$---is equal to \[ \frac1{k_n} \sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} \Bigl({\cal L}^n(\widetilde u^n_j; c_{(i-1)\Delta_n},\rho_{(i-1)\Delta_n})-{\cal L}^n(\widetilde u^n_j; c_{(i-p_n-1)\Delta_n},\rho_{(i-p_n-1)\Delta_n})\Bigr)\] or \begin{align*} &\frac{1}{k_n} \sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} \biggl( \int_{(i-p_n-1)\Delta_n}^{(i-1)\Delta_n} \tfrac{\partial}{\partial c} {\cal L}^n(\widetilde u^n_j;c_s,\rho_s) ({\sigma}_s^cdW_s + \overline {\sigma}^c_sd\overline W_s ) \\ &\quad+ \int_{(i-p_n-1)\Delta_n}^{(i-1)\Delta_n} \tfrac{\partial}{\partial \rho} {\cal L}^n(\widetilde u^n_j;c_s,\rho_s) ({\sigma}_s^\rho dW_s + \widetilde {\sigma}^\rho_sd\widetilde W_s )\\ &\quad + \iint_{(i-p_n-1)\Delta_n}^{(i-1)\Delta_n} \Bigl({\cal L}^n(\widetilde u^n_j;c_{s-}+{\gamma}^c(s,z),\rho_{s-}+{\gamma}^\rho(s,z))\\ &\qquad\qquad\qquad\qquad\qquad\qquad-{\cal L}^n(\widetilde u^n_j;c_{s-} ,\rho_{s-} ) \Bigr)(\mu-\nu)(ds,dz) \biggr), \end{align*} where the second step follows from It\^o's formula and by ignoring all drift terms of order $o'_p(\sqrt{p_n\Delta_n})$. At the same time, because $\Delta_n^{\varpi-1/2}=o'(\sqrt{p_n\Delta_n})$ by ((ref)) for all $\varpi>\frac34$ and $c$, $\Delta_n{\varphi}(u_n)$ and $\rho$ (if $\varpi\leq \frac34$) are It\^o semimartingales with bounded coefficients, we have $ e^{i(\widetilde u^n_j)_n \Delta^n_i y} = e^{\Delta^n_i\Theta((\widetilde u^n_j)_n)+i(\widetilde u^n_j)_n\Delta^n_i {\epsilon}}Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n} =e^{\Delta\overline\Theta((\widetilde u^n_j)_n)^n_{(2j-2)p_n}+i(\widetilde u^n_j)_n \Delta^n_{i,(2j-2)p_n}{\epsilon}}\times Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n} +o'_p(\sqrt{p_n\Delta_n})$, where $\Delta\overline \Theta(u)^n_i = -\frac12 u^2c_{i\Delta_n}\Delta_n + \Delta_n{\varphi}(u)_{i\Delta_n}$. Hence, \begin{align*} \widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)&= \frac1{k_n} \sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} \Bigl\{ e^{\Delta\Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}+i(\widetilde u^n_j)_n \Delta^n_{i,[2j-2]^m_n}{\epsilon}} (Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}-1)\\ &\quad+e^{\Delta\Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}(e^{i(\widetilde u^n_j)_n \Delta^n_{i,[2j-2]^m_n}{\epsilon}} -\Psi((\widetilde u^n_j)_n\Delta_n^\varpi \rho_{[2j-2]^m_n\Delta_n}))\Bigr\} \\ &\quad+o'_p(\sqrt{p_n\Delta_n}). \end{align*} By conditioning on ${\cal F}_\infty\vee{\cal G}_{[2j-2]^m_n}$, it is easy to see that the second part does not contribute to $\overline\mathbb{E}^n_{[2j-2]^m_n}[\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)\Delta^n_{2j} \widetilde L^{\mathrm{s}}_1(\widetilde u^n_j)]$. By definition, we have $Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}-1=\int_{(i-1)\Delta_n}^{i\Delta_n} Z^n_i((\widetilde u^n_j)_n)_{s}d{\Lambda}((\widetilde u^n_j)_n)_s$, where $${\Lambda}(u)_t =iu\int_0^t {\sigma}_r dW_r + \iint_0^t (e^{iu({\gamma}(r,z)+{\Gamma}(r,z))}-1) (\mu-\nu)(dr,dz),$$ so a straightforward computation shows that \begin{equation}\begin{split} & \overline\mathbb{E}^n_{[2j-2]^m_n}[\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)\Delta^n_{2j} \widetilde L^{\mathrm{s}}_1(\widetilde u^n_j)]\\ &\quad=\frac{1}{k_n^2} \sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} q^n_ie^{\Delta\Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}\Psi((\widetilde u^n_j)_n \Delta_n^\varpi\rho_{[2j-2]^m_n\Delta_n})\\ & \qquad \times\biggl( \iint_{(i-1)\Delta_n}^{i\Delta_n} \mathbb{E}^n_{[2j-2]^m_n}\Bigl[ (\cdots)Z^n_i((\widetilde u^n_j)_n)_s (e^{i(\widetilde u^n_j)_n({\gamma}(s,z)+{\Gamma}(s,z))}-1) \Bigr]{\lambda}(dz)ds\\ &\qquad+ \int_{(i-1)\Delta_n}^{i\Delta_n} i(\widetilde u^n_j)_n\mathbb{E}^n_{[2j-2]^m_n}\Bigl[ Z^n_i((\widetilde u^n_j)_n)_s{\sigma}_s\bigl(\tfrac{\partial}{\partial c} {\cal L}^n(\widetilde u^n_j;c_s,\rho_s) {\sigma}_s^c\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \tfrac{\partial}{\partial \rho} {\cal L}^n(\widetilde u^n_j;c_s,\rho_s) {\sigma}_s^\rho\bigr)\Bigr] ds\biggr)+o'_p(\sqrt{p_n\Delta_n}), \end{split} \end{equation} where $q^n_i$ are integers satisfying $0\leq q^n_i \leq p_n$ and $(\cdots)$ stands for the integrand of the $(\mu-\nu)$-integral in the expansion of $\Delta^n_{2j}\widetilde L^{\rm s}_1(\widetilde u^n_j)$ above. After a moment of thought, one realizes that $\lvert (\cdots)\rvert \leq CJ(z)$, where $C$ does not depend on $n$, ${\omega}$, $i$ or $j$. Since $\lvert Z^n_i((\widetilde u^n_j)_n)_s\rvert\leq C$ (as we have shown right below (ref)), it follows by dominated convergence that the ${\lambda}(dz)ds$-integral in the display above is $o'(\sqrt{\Delta_n})$. Since the $ds$-integral in (ref) is of order $\Delta_n^{1-(\varpi\wedge1/2)}$, we only need to further consider the case where $\varpi\geq\frac12$. Then the $ds$-integral is of size $O(\sqrt{\Delta_n})$, which implies that we are free to make any additional modification of order $o'(1)$. Because of the $L^2$-continuity properties of ${\sigma}$ and ${\sigma}^c$, and of $\rho$ and ${\sigma}^\rho$ if $\varpi\leq \frac34$, and because $ \mathbb{E}^n_{[2j-2]^m_n}[Z^n_i((\widetilde u^n_j)_n)_s] = 1$, it suffices, instead of (ref), to consider \begin{equation} \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Re\Biggl\{ \frac{\Delta^n_{2j-2}\widetilde L^{\mathrm{v}}(\widetilde u^n_{j-1})}{{\frak L}(Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})}\Biggr\}\Re\Biggl\{\frac{ \beta^n_j}{({\frak L}(Y^n(\widetilde u^n_j)_{[2j-2]^m_n\Delta_n}))^2}\Biggr\} \end{equation} where \begin{align*} \beta^n_j&=i\widetilde u^n_j \sqrt{\Delta_n}{\sigma}_{[2j-4]^m_n\Delta_n}(\tfrac{\partial}{\partial c} {\cal L}^n(\widetilde u^n_j;c_{[2j-4]^m_n\Delta_n},\rho_{[2j-4]^m_n\Delta_n}) {\sigma}_{[2j-4]^m_n\Delta_n}^c\\ &\quad+ \tfrac{\partial}{\partial \rho} {\cal L}^n(\widetilde u^n_j;c_{[2j-4]^m_n\Delta_n},\rho_{[2j-4]^m_n\Delta_n}) {\sigma}_{[2j-4]^m_n\Delta_n}^\rho ). \end{align*} In fact, we can also replace $Y^n(\widetilde u^n_j)_{[2j-2]^m_n\Delta_n}$ in (ref) first by $$\exp(-\frac12(\widetilde u^n_j)^2c_{[2j-2]^m_n\Delta_n}\Delta_n^{(1-2\varpi)_+})\Psi(\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+}\rho_{[2j-2]^m_n\Delta_n})$$ and then, in a second step, by $$\exp(-\frac12(\widetilde u^n_j)^2c_{[2j-4]^m_n\Delta_n}\Delta_n^{(1-2\varpi)_+})\Psi(\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+}\rho_{[2j-4]^m_n\Delta_n}).$$ Thanks to ((ref)) and ((ref)), we can also substitute $\widetilde u^n_{j-1}$ for $\widetilde u^n_j$ in the last expression and in the definition of $\beta^n_j$. After all these modifications, the $j$th term in (ref) will have a zero ${\cal H}^n_{[2j-4]^m_n}$-conditional expectation, so a martingale argument shows that (ref) is asymptotically negligible. Hence, in (ref), we can choose $ \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}} = \Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm II}-\Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm III}-\Delta^n_{2j} \overline {\frak c}(\widetilde u^n_j)_{\rm{s}}^{\rm V}$. The next reduction is to remove $\Delta^n_{2j-2\ell} \overline {\frak c}(\widetilde u^n_{j-\ell})_{\rm{s}}^{\rm III}+\Delta^n_{2j-2\ell} \overline {\frak c}(\widetilde u^n_{j-\ell})_{\rm{s}}^{\rm V}$ ($\ell\in\{0,1\}$), for which we have to show that \begin{align*} &\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} (\Delta^n_{2j-2 } \overline {\frak c}(\widetilde u^n_{j-1})_{\rm{s}}^{\rm III}+\Delta^n_{2j-2 } \overline {\frak c}(\widetilde u^n_{j-1})_{\rm{s}}^{\rm V})\Re\Biggl\{ \frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}}(\widetilde u^n_j)}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-1}(\widetilde u^n_j))}\Biggr\}\\ &\quad+\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Delta^n_{2j-2}\overline {\frak c}(\widetilde u^n_{j-1})_{\rm s}(\Delta^n_{2j} \overline {\frak c}(\widetilde u^n_{j})_{\rm{s}}^{\rm III}+\Delta^n_{2j} \overline {\frak c}(\widetilde u^n_{j})_{\rm{s}}^{\rm V}) \end{align*} is asymptotically negligible. As before, we can replace $\widetilde L^{n,\mathrm{s}}_{2k-\ell}(\widetilde u^n_k)$ by $Y^n(\widetilde u^n_k)_{[2k-2]^m_n\Delta_n}$ ($k\in\{j-1,j\}$, $\ell\in\{0,1\}$). By a martingale argument, we can further take conditional expectation with respect to ${\cal H}^n_{[2j-2]^m_n}$, which eliminates the first part of the sum above, while it renders the second part negligible by (ref) and (ref). This shows that we can choose $ \Delta^n_{2j} \overline c(\widetilde u^n_j)_{\rm{s}} = \Delta^n_{2j} \overline c(\widetilde u^n_j)_{\rm{s}}^{\rm II} $. Lastly, we want to show that we can replace $\overline V^{\prime n}_{\rm s}$ as defined in (ref) (with $ \Delta^n_{2j} \overline c(\widetilde u^n_j)_{\rm{s}} = \Delta^n_{2j} \overline c(\widetilde u^n_j)_{\rm{s}}^{\rm II}$) by $\widetilde V^{\prime n}$ from (ref), which will conclude the proof. We proceed in three steps by decomposing $\overline V^{\prime n}_{\rm s}-\widetilde V^{\prime n}=D^n_2 + D^n_3+D^n_4$, where (recall the definition of ${\cal L}^n_{j}(u)$ from the statement of the lemma) \begin{align*} D^n_2 &=\overline V^{\prime n}_{\rm s}- \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Re\Biggl\{\frac{ \Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}(\overline Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})}\Biggr\} \Re\Biggl\{\frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_j)}{{\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\Biggr\},\\ D^n_3&=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n} \Re\Biggl\{\frac{ \Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}(\overline Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})}\Biggr\}\Re\Biggl\{\frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j})}{{\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})} -\frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j})}{{\frak L}({\cal L}^n_{j}(\widetilde u^n_{j}))}\Biggr\} ,\\ D^n_4 &= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\Re\Biggl\{\frac{ \Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_j)}{{\frak L}({\cal L}^n_{j}(\widetilde u^n_{j}))}\Biggr\}\Re\Biggl\{\frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}(\overline Y^n(\widetilde u^n_{j-1})_{[2j-4]^m_n\Delta_n})} -\frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1}))}\Biggr\} \end{align*} and $\overline Y^n(u)_t$ was defined before (ref). Because $D^n_3$ is a martingale sum and we have $\lvert \overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}-{\cal L}^n_{j}(\widetilde u^n_{j})\rvert =o'(1)$ by (ref) and (ref), we readily obtain $\mathbb{E}[\lvert D^n_3\rvert^2]\to0$. The same argument shows that $\mathbb{E}[\lvert D^n_4\rvert^2]\to0$. Regarding $D^n_2$, we use the expansion $1/{\frak L}(z)=1/{\frak L}(z_0)-( (z-z_0)\log \lvert z_0\rvert+z_0\Re\{ (z-z_0)/z_0\})/({\frak L}(z_0))^2 + O(\lvert z-z_0\rvert^2/(\lvert {\frak L}(z)\rvert\wedge \lvert{\frak L}(z_0)\rvert)^3)$ to bound \begin{align*} &\overline\mathbb{E}^n_{[2j-2]^m_n}\Biggl[ \biggl\lvert\frac{1}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-1}(\widetilde u^n_j))}-\frac{1}{{\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\\ &\quad+\frac{\log \lvert\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}\rvert }{({\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}))^2k_n} \sum_{i=(2j-2)p_n+1}^{(2j-2)p_n+k_n} \Bigl(\overline Y^n(\widetilde u^n_{j})_{(i-1)\Delta_n}-\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}\Bigr) \\ &\quad+\frac{ 1 }{{\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})k_n} \sum_{i=(2j-2)p_n+1}^{(2j-2)p_n+k_n} \Re\biggl\{\frac{\overline Y^n(\widetilde u^n_{j})_{(i-1)\Delta_n}-\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}}{{\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\biggr\}\biggr\rvert\Biggr] \end{align*} by ${Cp_n\Delta_n}/{\theta_n^6}$. Since we have $ \overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}={\cal L}^n_{j}(\widetilde u^n_{j})+o'(1)$ and $\overline Y^n(\widetilde u^n_{j})_{(i-1)\Delta_n}-\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n}=\psi^{n,j}_i+o'(\sqrt{p_n\Delta_n})$, where $ \psi^{n,j}_i=\tfrac{\partial}{\partial c} {\cal L}^n(\widetilde u^n_j;c_{[2j-2]^m_n\Delta_n},\rho_{[2j-2]^m_n\Delta_n}) \linebreak (c_{(i-1)\Delta_n}-c_{[2j-2]^m_n\Delta_n}) + \tfrac{\partial}{\partial \rho} {\cal L}^n(\widetilde u^n_j;c_{[2j-2]^m_n\Delta_n},\rho_{[2j-2]^m_n\Delta_n}) (\rho_{(i-1)\Delta_n}-\rho_{[2j-2]^m_n\Delta_n})$, the previous bound implies that instead of $D^n_2$, it suffices to study \begin{align*} D^n_{21}&= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}} \sum_{j\in{\cal J}_n}\Re\Biggl\{\frac{ \Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})}{{\frak L}(\widetilde L^{n,\mathrm{s}}_{2j-3}(\widetilde u^n_{j-1}))}\Biggr\}\Biggl[\Re\Biggl\{ \frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_j)\log \lvert {\cal L}^n_{j}(\widetilde u^n_{j})\rvert}{({\frak L}( {\cal L}^n_{j}(\widetilde u^n_{j})))^2k_n} \sum_{i=(2j-2)p_n+1}^{(2j-2)p_n+k_n} \psi^{n,j}_i\Biggr\}\\ &\quad +\Re\Biggl\{ \frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_j) }{{\frak L}( {\cal L}^n_{j}(\widetilde u^n_{j})) } \Biggr\}\frac1{k_n}\sum_{i=(2j-2)p_n+1}^{(2j-2)p_n+k_n} \Re\biggl\{\frac{\psi^{n,j}_i}{{\frak L}({\cal L}^n_{j}(\widetilde u^n_{j}))}\biggr\}\Biggr] \end{align*} and \begin{align*} D^n_{22}&= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}} \sum_{j\in{\cal J}_n}\Biggl[\Re\Biggl\{\frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1})\log \lvert {\cal L}^n_{j-1}(\widetilde u^n_{j-1})\rvert}{({\frak L}( {\cal L}^n_{j-1}(\widetilde u^n_{j-1})))^2k_n}\sum_{i=(2j-4)p_n+1}^{(2j-4)p_n+k_n} \psi^{n,j-1}_i\Biggr\}\\ &\quad +\Re\Biggl\{ \frac{\Delta^n_{2j-2} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j-1}) }{{\frak L}( {\cal L}^n_{j-1}(\widetilde u^n_{j-1}))} \Biggr\}\frac1{k_n}\sum_{i=(2j-4)p_n+1}^{(2j-4)p_n+k_n} \Re\biggl\{\frac{\psi^{n,j-1}_i}{{\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1}))}\biggr\}\Biggr]\\ &\quad\times\Re\Biggl\{ \frac{\Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_j)}{{\frak L}(\overline Y^n(\widetilde u^n_{j})_{[2j-2]^m_n\Delta_n})}\Biggr\}. \end{align*} Since the $j$th term in $D^n_{22}$ has a zero ${\cal H}^n_{[2j-2]^m_n}$-conditional expectation, we can argue as in the analysis of $D^n_3$ to show that $D^n_{22}$ converges to $0$ in $L^2$. For $D^n_{21}$, by the same argument, we can take conditional expectation with respect to ${\cal H}^n_{[2j-2]^m_n}$ for the $j$th term, which can be computed similarly to (ref). The final result after a tedious computation is that $D^n_{21}$ vanishes asymptotically. \end{proof} \begin{proof}[Proof of Lemma (ref)] Let $\zeta^{n,\rm v}_{j} = \Re \{ { \Delta^n_{2j} \widetilde L^{\mathrm{v}} (\widetilde u^n_{j})}/{{\frak L}({\cal L}^n_j(\widetilde u^n_j))} \}$. Then we can decompose $\widetilde V^n = \widetilde V^{n}_1+ \widetilde V^{n}_2 - \widetilde V^{n}_3-\widetilde V^{n}_4$, where \begin{align*} \widetilde V^{n}_1&=\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\zeta^{n,\rm v}_{j-1} \zeta^{n,\rm v}_{j}, & \widetilde V^{n}_2& =\frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\zeta^{n,\rm v}_{j-1-\mathds{1}/2} \zeta^{n,\rm v}_{j-\mathds{1}/2}, \\ \widetilde V^{n}_3&= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\zeta^{n,\rm v}_{j-1-\mathds{1}/2} \zeta^{n,\rm v}_{j},& \widetilde V^{n}_4&= \frac{k_n}{\sqrt{\lvert {\cal J}_n\rvert}}\sum_{j\in{\cal J}_n}\zeta^{n,\rm v}_{j-\mathds{1}/2} \zeta^{n,\rm v}_{j-1} \end{align*} In each of the five sums, the $j$th term is ${\cal H}^n_{(2j-1)p_n+k_n }$-measurable with a zero ${\cal H}^n_{[2j-2]^m_n }$-conditional expectation. Therefore, $(\widetilde V^n_1,\dots,\widetilde V^n_4)$ is a four-dimensional martingale array and we can use Theorem 2.2.15 in JP12 to prove its convergence, from which ((ref)) can be easily deduced. Verifying the assumptions of Theorem 2.2.15 in JP12 is straightforward, so we only derive the asymptotic covariance and leave the remaining conditions to the reader. A moment's thought reveals that the limits of $\widetilde V^n_1,\dots,\widetilde V^n_4$ are ${\cal F}_\infty$-conditionally independent, so all we are left to show is that their ${\cal F}_\infty$-conditional variances are given, respectively, by $\frac{1}{T}\int_{I_T} q_t^2dt$, $\frac{1}{T}\int_{I_T} q_{t-1}^2dt$ and, for both $\widetilde V^n_3$ and $\widetilde V^n_4$, by $\frac{1}{T}\int_{I_T} q_tq_{t-1} dt$. As the proof is almost identical, we only determine the ${\cal F}_\infty$-conditional variance of $\widetilde V^n_1$, which we denote by $Q^{(1)}$ and is given by the limit as $n\to\infty$ of \begin{equation} Q_n= \frac{k_n^2}{ \lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n} \overline\mathbb{E}^n_{[2j-2]^m_n}[(\zeta^{n,\rm v}_{j-1} \zeta^{n,\rm v}_{j})^2] =\frac{k_n^2}{ \lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n} (\zeta^{n,\rm v}_{j-1} )^2\overline\mathbb{E}^n_{[2j-2]^m_n} [ ( \zeta^{n,\rm v}_{j})^2 ].\end{equation} By a martingale argument,\footnote{The difference between (ref) and (ref) can be split into two martingale sums, one summing over even and one summing over odd values of $j$.} this is asymptotically equivalent to \begin{equation} \frac{k_n^2}{ \lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n} \overline\mathbb{E}^n_{[2j-4]^m_n} [ (\zeta^{n,\rm v}_{j-1} )^2\overline\mathbb{E}^n_{[2j-2]^m_n} [ ( \zeta^{n,\rm v}_{j})^2 ]]. \end{equation} To determine the limit of (ref), we start with the following a priori estimate, which follows from the bounds in (ref):\footnote{Note that one can replace $e^{iU_n\Delta^n_i y}$ in $\xi^n_i$ by $1-e^{iU_n\Delta^n_i y}$ without changing anything, where $U=\widetilde u^n_j$ in our case. Then the claim follows from the bound $\overline\mathbb{E}^n_{[2j-2]^m_n}[\lvert 1-e^{i(\widetilde u^n_j)_n\Delta^n_i y}\rvert^2]\leq (\widetilde u^n_j)^2_n\overline \mathbb{E}^n_{[2j-2]^m_n}[(\Delta^n_i y)^2]\leq C(\widetilde u^n_j)^2\leq C\theta_n^2$.} \begin{equation} \overline\mathbb{E}^n_{[2j-2]^m_n}[\lvert \Delta^n_{2j}\widetilde L^{\rm v}(\widetilde u^n_j)\rvert^2]= \mathbb{E}^n_{[2j-2]^m_n}[\lvert \Delta^n_{2j}\widetilde L^{\rm v}(\widetilde u^n_j)\rvert^2] \leq C \theta_n^2/k_n, \end{equation} where $C$ does not depend on ${\omega}$ or $j$. Moreover, $\widetilde L^{n,\mathrm{v}}_{2j}(\widetilde u^n_j)$ and $\widetilde L^{n,\mathrm{v}}_{2j-1}(\widetilde u^n_j)$ are uncorrelated and ${\cal L}^n_j(\widetilde u^n_j)$ is ${\cal H}^n_{[2j-2]^m_n}$-measurable, so \begin{equation*} \overline\mathbb{E}^n_{[2j-2]^m_n} [ ( \zeta^{n,\rm v}_{j})^2 ]=\overline\mathbb{E}^n_{[2j-2]^m_n}\biggl[\biggl( \Re\biggl\{\frac{ \widetilde L^{n,\mathrm{v}}_{2j} ( \widetilde u^n_j)}{{\frak L}({\cal L}^n_j(\widetilde u^n_j))}\biggr\}\biggr)^2+\biggl( \Re\biggl\{\frac{ \widetilde L^{n,\mathrm{v}}_{2j-1} ( \widetilde u^n_j)}{{\frak L}({\cal L}^n_j(\widetilde u^n_j))}\biggr\}\biggr)^2\biggr]. \end{equation*} For $\ell\in\{0,1\}$, we have the identity \begin{align}\nonumber \biggl( \Re\biggl\{\frac{ \widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j)}{{\frak L}({\cal L}^n_j(\widetilde u^n_j))}\biggr\}\biggr)^2 &=\frac{(\Re\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) )^2(\Re {\frak L}({\cal L}^n_j(\widetilde u^n_j)))^2}{\lvert {\frak L}({\cal L}^n_j(\widetilde u^n_j))\rvert^4}+ \frac{(\Im\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) )^2(\Im {\frak L}({\cal L}^n_j(\widetilde u^n_j)))^2}{\lvert{\frak L}( {\cal L}^n_j(\widetilde u^n_j))\rvert^4}\\ & \quad+\frac{2(\Re\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) )(\Re {\frak L}({\cal L}^n_j(\widetilde u^n_j)))(\Im\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) )(\Im {\frak L}({\cal L}^n_j(\widetilde u^n_j)))}{\lvert {\frak L}({\cal L}^n_j(\widetilde u^n_j))\rvert^4}. \end{align} Since $\lvert \cos x-1 \rvert \leq \frac12 x^2$ and $\lvert \sin x \rvert \leq x$, one can argue as in (ref) to derive the bounds \begin{equation} \overline\mathbb{E}^n_{[2j-2]^m_n} [ \lvert\Re \widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) \rvert^2 ]\leq C\theta_n^4/k_n,\qquad \overline\mathbb{E}^n_{[2j-2]^m_n} [ \lvert\Im \widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) \rvert^2 ]\leq C\theta_n^2/k_n. \end{equation} For ${\frak L}({\cal L}^n_j(\widetilde u^n_j))$, there are $C,C_1,C_2>0$ such that \begin{equation} C_1\theta_n^2\leq \lvert \Re {\frak L}({\cal L}^n_j(\widetilde u^n_j))\rvert, \lvert{\frak L}({\cal L}^n_j(\widetilde u^n_j))\rvert\leq C_2\theta_n^2,\quad \lvert\Im{\frak L}({\cal L}^n_j(\widetilde u^n_j))\rvert \leq C \theta_n^5\Delta_n^{3(\varpi-1/2)_+}, \end{equation} where the first set of bounds follows as in Lemma (ref) and the last bound holds because $\lvert e^{ix}-1-ix+\frac12 x^2\rvert\leq C\lvert x\rvert^3$ and $\mathbb{E}[\Delta\chi_1]=0$. In conjunction with (ref), these bounds show that the second term on the right-hand side of (ref) is negligible for computing the limit of $Q_n$. Regarding the last term in (ref), note that (ref) and the Cauchy--Schwarz inequality imply $\overline\mathbb{E}^n_{[2j-2]^m_n} [(\Re\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j))(\Im\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j)) ]\leq C\theta_n^3/k_n$. Therefore, $\overline\mathbb{E}^n_{[2j-2]^m_n}$ of the last term in (ref) is bounded by $C\theta_n^2/k_n$ and does not contribute to $Q^{(1)}$. Thus, $Q_n$ is asymptotically equivalent to \begin{align*} & \frac{k_n^2}{ \lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n}\overline\mathbb{E}^n_{[2j-4]^m_n}\Biggl[\frac{[(\Re\widetilde L^{n,\mathrm{v}}_{2j-2} ( \widetilde u^n_{j-1}) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-3} ( \widetilde u^n_{j-1}) )^2](\Re {\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1})))^2}{\lvert {\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1}))\rvert^4}\\ &\qquad\qquad\qquad \qquad\times\mathbb{E}^n_{[2j-2]^m_n}\biggl[\frac{[(\Re\widetilde L^{n,\mathrm{v}}_{2j} ( \widetilde u^n_j) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-1} ( \widetilde u^n_j) )^2](\Re{\frak L}( {\cal L}^n_j(\widetilde u^n_j)))^2}{\lvert{\frak L}( {\cal L}^n_j(\widetilde u^n_j))\rvert^4}\biggr]\Biggr]. \end{align*} In fact, by (ref) and (ref), the previous display is $O_p(1)$, so we are allowed to make any further $o(1)$-modification. For example, we can ignore the imaginary part of ${\cal L}^n_{j-\ell}(\widetilde u^n_{j-\ell})$ ($\ell=0,1$) and replace ${\cal L}^n_{j}(\widetilde u^n_{j})$ by ${\cal L}^n_{j-1}(\widetilde u^n_{j-1})$. This shows that $Q_n$ is asymptotically equivalent to \begin{equation}\begin{split} & \frac{k_n^2}{ \lvert {\cal J}_n\rvert}\sum_{j\in{\cal J}_n} \frac{1}{(\Re{\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1})))^4} \overline\mathbb{E}^n_{[2j-4]^m_n}\Bigl[ \bigl[(\Re\widetilde L^{n,\mathrm{v}}_{2j-2} ( \widetilde u^n_{j-1}) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-3} ( \widetilde u^n_{j-1}) )^2\bigr] \\ &\qquad\qquad\qquad\qquad\qquad \qquad\qquad \times\overline\mathbb{E}^n_{[2j-2]^m_n} [ (\Re\widetilde L^{n,\mathrm{v}}_{2j} ( \widetilde u^n_j) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-1} ( \widetilde u^n_j) )^2 ]\Bigr]. \end{split}\end{equation} Regarding the denominator, we have \begin{equation} \begin{split} \Re{\frak L}({\cal L}^n_{j-1}(\widetilde u^n_{j-1}))&=-\tfrac12(\widetilde u^n_j)^2(c_{[2j-2]^m_n\Delta_n}\mathbf 1_{\{\varpi\geq\frac12\}}+\rho^2_{[2j-2]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^2]\mathbf 1_{\{\varpi\leq\frac12\}})\\ &\quad\times\exp(-\tfrac12(\theta_0^2/\widetilde\eta^n_j)^2c_{[2j-2]^m_n\Delta_n})+o(\theta_n^2).\end{split} \end{equation} Moreover, since $(\chi_i)_{i\in\mathbb{Z}}$ is $m$-dependent and $\cos(x)\cos(y)=\frac12(\cos(x+y)+\cos(x-y))$, \begin{equation}\begin{split} &\overline\mathbb{E}^n_{[2j-2]^m_n} [ (\Re\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) )^2]= \mathbb{E}^n_{[2j-2]^m_n} [ (\Re\widetilde L^{n,\mathrm{v}}_{2j-\ell} ( \widetilde u^n_j) )^2]\\ & \quad =\frac1{k_n^2}\sum_{i,i'} \mathbb{E}^n_{[2j-2]^m_n} \Bigl[ \cos((\widetilde u^n_j)_n\Delta^n_iy)\cos((\widetilde u^n_j)_n\Delta^n_{i'}y)\\ &\qquad\qquad\qquad\qquad\qquad\qquad-\mathbb{E}^n_{i-1}[\cos((\widetilde u^n_j)_n\Delta^n_iy)]\mathbb{E}^n_{i'-1}[\cos((\widetilde u^n_j)_n\Delta^n_{i'}y)] \Bigr]\\ & \quad=\frac1{2k_n^2}\sum_{i,i'} \mathbb{E}^n_{[2j-2]^m_n} \Bigl[ \mathbb{E}^n_{i\wedge i'-1}[\cos((\widetilde u^n_j)_n(\Delta^n_i y+\Delta^n_{i'}y))]\\ &\qquad+ \mathbb{E}^n_{i\wedge i'-1}[\cos((\widetilde u^n_j)_n(\Delta^n_i y-\Delta^n_{i'}y))] -2\mathbb{E}^n_{i-1}[\cos((\widetilde u^n_j)_n\Delta^n_i y) ]\mathbb{E}^n_{i'-1}[\cos((\widetilde u^n_j)_n\Delta^n_{i'} y) ]\Bigr] \end{split}\end{equation} for $\ell\in\{0,1\}$, where the sum ranges over all $i,i'\in\{(2j-\ell-1)p_n+1,\dots,(2j-\ell-1)p_n+k_n\}$ such that $i-i'\in\{-m-1,\dots,m+1\}$. Using (ref) for the second step, (ref) for the fourth one and ((ref)) and ((ref)) for the last one, we have \begin{align*} &\mathbb{E}^n_{i-1}[\cos((\widetilde u^n_j)_n\Delta^n_i y) ] =\Re\mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_i y} ]\\ &\quad=\Re\mathbb{E}^n_{i-1}[e^{\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}e^{i(\widetilde u^n_j)_n\Delta^n_{i,[2j-2]^m_n} {\epsilon}}]+O(p_n\Delta_n) \\ &\quad=\Re\mathbb{E}^n_{i-1}[e^{\Delta \Theta((\widetilde u^n_j)_n)^n_{[2j-2]^m_n}}Z^n_i((\widetilde u^n_j)_n)_{i\Delta_n}\Psi((\widetilde u^n_j)_n\Delta_n^\varpi\rho_{[2j-2]^m_n\Delta_n})]+O(p_n\Delta_n) \\ &\quad=\exp(-\tfrac12(\widetilde u^n_j)^2c_{[2j-2]^m_n\Delta_n}\Delta_n^{(1-2\varpi)_+})\Re \Psi(\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+}\rho_{[2j-2]^m_n\Delta_n})+o'(1)\\ &\quad=\exp(-\tfrac12(\widetilde u^n_{j-1})^2c_{[2j-4]^m_n\Delta_n}\Delta_n^{(1-2\varpi)_+})\Re \Psi(\widetilde u^n_{j-1} \Delta_n^{(\varpi-1/2)_+}\rho_{[2j-4]^m_n\Delta_n})+o'(1). \end{align*} Similarly, with the notations $\Psi^\pm_r(u)=\mathbb{E}[e^{iu(\Delta\chi_{r+1}\pm\Delta\chi_1)}]$ and $r=i-i'$, we can deduce \begin{align*} \mathbb{E}^n_{i\wedge i'-1}[\cos((\widetilde u^n_j)_n(\Delta^n_i y\pm\Delta^n_{i'}y))] &=\exp(- (1\pm\mathbf 1_{\{r=0\}})(\widetilde u^n_{j-1})^2c_{[2j-4]^m_n\Delta_n}\Delta_n^{(1-2\varpi)_+}) \\ &\quad\times \Re \Psi^\pm_r(\widetilde u^n_{j-1} \Delta_n^{(\varpi-1/2)_+}\rho_{[2j-4]^m_n\Delta_n})+o'(1). \end{align*} Now, if $\varpi>\frac12$, then the contributions of $\Psi$ and $\Psi^\pm$ are $o'(1)$ and only the terms where $r=0$ remain, which via (ref) shows that for $\ell\in\{0,1\}$, \begin{align*} & \overline\mathbb{E}^n_{[2j-2\ell-2]^m_n} [ (\Re\widetilde L^{n,\mathrm{v}}_{2j-2\ell} ( \widetilde u^n_{j-\ell}) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-2\ell-1} ( \widetilde u^n_{j-\ell}) )^2 ]\\ &\quad=\frac{(1-\exp(-(\widetilde u^n_{j-1})^2c_{[2j-4]^m_n\Delta_n}))^2}{k_n} + o'(1). \end{align*} If $\varpi<\frac12$, the contributions of $\Psi$ and $\Psi^\pm$ dominate asymptotically. Moreover, we have $\theta_n\to0$ by assumption, so expanding $\cos(x)=1-\frac12 x^2+\frac1{24}x^4+O(x^6)$, we get \begin{align*} &\mathbb{E}^n_{i-1}[\cos((\widetilde u^n_j)_n\Delta^n_i y) ]=\Re \Psi(\widetilde u^n_{j-1} \rho_{[2j-4]^m_n\Delta_n})+o'(1) \\ &\quad=1-\tfrac12(\widetilde u^n_{j-1})^2\rho_{[2j-4]^m_n\Delta_n}^2 \mathbb{E}[(\Delta\chi_1)^2] + \tfrac1{24}(\widetilde u^n_{j-1})^4\rho_{[2j-4]^m_n\Delta_n}^4\mathbb{E}[(\Delta\chi_1)^4]+O(\theta_n^6) \end{align*} and \begin{align*} \mathbb{E}^n_{i\wedge i'-1}[\cos((\widetilde u^n_j)_n(\Delta^n_i y\pm\Delta^n_{i'}y))] &= \Re \Psi^\pm_r(\widetilde u^n_{j-1} \rho_{[2j-4]^m_n\Delta_n})+o'(1)\\ & =1-\tfrac12(\widetilde u^n_{j-1})^2\rho_{[2j-4]^m_n\Delta_n}^2 \mathbb{E}[(\Delta\chi_{r+1}\pm\Delta\chi_1)^2] \\ &\quad+ \tfrac1{24}(\widetilde u^n_{j-1})^4\rho_{[2j-4]^m_n\Delta_n}^4\mathbb{E}[(\Delta\chi_{r+1}\pm\Delta\chi_1)^4]+O(\theta_n^6), \end{align*} which inserted in (ref) and after simplifications yields \begin{align*} &\overline\mathbb{E}^n_{[2j-2\ell-2]^m_n} [ (\Re\widetilde L^{n,\mathrm{v}}_{2j-2\ell} ( \widetilde u^n_{j-\ell}) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-2\ell-1} ( \widetilde u^n_{j-\ell}) )^2 ]\\ &\quad=\frac{(\widetilde u^n_{j-1})^4}{2k_n}\rho^4_{[2j-4]^m_n\Delta_n}\biggl(\mathbf{Var}((\Delta \chi_1)^2)+2\sum_{r=1}^{m+1}\mathbf{Cov}((\Delta\chi_{r+1})^2,(\Delta\chi_1)^2) \biggr)+ O(\theta_n^6). \end{align*} If $\varpi=\frac12$, we also have $\theta_n\to0$, so combining the previous results, we obtain \begin{align*} \mathbb{E}^n_{i-1}[\cos((\widetilde u^n_j)_n\Delta^n_i y) ]&=1-\tfrac12( \widetilde u^n_{j-1} )^2 (c_{[2j-4]^m_n\Delta_n}+\rho^2_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^2])\\ & \quad +\tfrac1{24}( \widetilde u^n_{j-1} )^4(3c_{[2j-4]^m_n\Delta_n}^2 +6c_{(2j-4)p_n\Delta_n}\rho^2_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^2]\\ &\quad+\rho^4_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^4]) +O(\theta_n^6) \end{align*} and \begin{align*} &\mathbb{E}^n_{i\wedge i'-1}[\cos((\widetilde u^n_j)_n(\Delta^n_i y\pm\Delta^n_{i'}y))]\\ & =1-( \widetilde u^n_{j-1} )^2\bigl((1\pm\mathbf 1_{\{r=0\}}) c_{[2j-4]^m_n\Delta_n}+\tfrac12\rho^2_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_{r+1}\pm\Delta\chi_1)^2]\bigr)\\ &\quad +( \widetilde u^n_{j-1} )^4\bigl(\tfrac12(1\pm\mathbf 1_{\{r=0\}})^2c_{[2j-4]^m_n\Delta_n}^2 +\tfrac1{24}\rho^4_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_{r+1}\pm\Delta\chi_1)^4]\\ &\quad+\tfrac12(1\pm\mathbf 1_{\{r=0\}})c_{[2j-4]^m_n\Delta_n}\rho^2_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_{r+1}\pm\Delta\chi_1)^2]\bigr) +O(\theta_n^6). \end{align*} Thus, if $\varpi=\frac12$, we have \begin{align*} &\overline\mathbb{E}^n_{[2j-2\ell-2]^m_n} [ (\Re\widetilde L^{n,\mathrm{v}}_{2j-2\ell} ( \widetilde u^n_{j-\ell}) )^2+(\Re\widetilde L^{n,\mathrm{v}}_{2j-2\ell-1} ( \widetilde u^n_{j-\ell}) )^2 ]\\ &\quad=\frac{(\widetilde u^n_{j-1})^4}{2k_n}\Biggl(2c^2_{[2j-4]^m_n\Delta_n}+4c_{[2j-4]^m_n\Delta_n}\rho^2_{[2j-4]^m_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^2]\\ &\qquad +\rho^4_{[2j-4]^m_n\Delta_n}\biggl(\mathbf{Var}((\Delta \chi_1)^2)+2\sum_{r=1}^m\mathbf{Cov}((\Delta\chi_{r+1})^2,(\Delta\chi_1)^2)\biggr)\Biggr) + O(\theta_n^6). \end{align*} Inserting the obtained expansions in (ref) and noting that $\mathbb{E}[\lvert (\widetilde u^n_{j-1})^2-\theta_n^2/\eta_{(2j-4)p_n\Delta_n}\rvert]\leq \theta_n^2(K\vee \eta_0^{-1})^2\mathbb{E}[\lvert \eta^n_j-\eta_{(2j-2)p_n\Delta_n}\rvert] =o(\theta_n^2)$ by ((ref)) and $\log \lvert \Psi(u)\rvert = \Re \Log \Psi(u)=\Re \Log (1-\frac12 u^2 \mathbb{E}[(\Delta\chi_1)^2]+O(u^3))=-\frac12u^2\mathbb{E}[(\Delta\chi_1)^2]+O(u^3)$, one can use classical Riemann approximation results to show $Q^{(1)}=\frac{1}{T}\int_{I_T} q_t^2 dt$ (by distinguishing the cases $\varpi>\frac12$, $\varpi=\frac12$ and $\varpi<\frac12$, and in the first case, further whether $\theta_0=0$ or not). \end{proof} \begin{proof}[Proof of Lemma (ref)] We decompose $W^n= W^{\prime n}+W^{\prime \prime n}$, where \begin{align*} W^{\prime n}&= \frac{ k_n^2 }{\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n}[(\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_{j}) )^2 +(\Delta^n_{2j-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-\mathds{1}/2}) )^2]\\ &\quad\times[(\Delta^n_{2j-2} \widetilde {\frak c}( \widetilde{u}^n_{j-1}) )^2 + (\Delta^n_{2j-2-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-1-\mathds{1}/2}) )^2],\\ W^{\prime\prime n}&=\frac{ 2k_n^2 }{\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n}\Bigl\{2\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_{j}) \Delta^n_{2j-2} \widetilde {\frak c}( \widetilde{u}^n_{j-1}) \Delta^n_{2j-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-\mathds{1}/2}) \Delta^n_{2j-2-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-1-\mathds{1}/2}) \\ &\quad -[(\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_{j}) )^2+(\Delta^n_{2j-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-\mathds{1}/2}) )^2]\Delta^n_{2j-2} \widetilde {\frak c}( \widetilde{u}^n_{j-1}) \Delta^n_{2j-2-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-1-\mathds{1}/2}) \\ &\quad -[(\Delta^n_{2j-2} \widetilde {\frak c}( \widetilde{u}^n_{j-1}) )^2+(\Delta^n_{2j-2-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-1-\mathds{1}/2}) )^2]\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_{j}) \Delta^n_{2j-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-\mathds{1}/2}) \Bigr\}. \end{align*} As seen in the proof of Lemma (ref), we have $\overline\mathbb{E}^n_{[2j-2]^m_n}[\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_{j})]=\overline\mathbb{E}^n_{[2j-2]^m_n}[\Delta^n_{2j} \overline {\frak c}( \widetilde{u}^n_{j})^{\rm I}_{\rm s}+\Delta^n_{2j} \overline {\frak c}( \widetilde{u}^n_{j})^{\rm II}_{\rm s}]+o'(1/\sqrt{k_n})=o'(1/\sqrt{k_n})$. Therefore, invoking a martingale argument, we can apply $\overline\mathbb{E}^n_{[2j-2]^m_n}$ to the $j$th term defining $W^{\prime\prime n}$, after which the only expression left to be analyzed is $ -\frac{ 2k_n^2 }{\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n}\overline\mathbb{E}^n_{[2j-2]^m_n}[(\Delta^n_{2j} \widetilde {\frak c}( \widetilde{u}^n_{j}) )^2]\Delta^n_{2j-2} \widetilde {\frak c}( \widetilde{u}^n_{j-1}) \Delta^n_{2j-2-\mathds{1}} \widetilde {\frak c}( \widetilde{u}^n_{j-1-\mathds{1}/2}) $. Similarly to how we eliminated $\Delta^n_{2j-2}\overline {\frak c}(\widetilde u^n_{j-1})^{\rm I}_{\rm s}$ in the paragraph following (ref), one can verify that this term converges in probability to $0$. Therefore, only $W^{\prime n}$ contributes asymptotically. By our analysis of $\Delta^n_{2j} \widetilde{{\frak c}}(\widetilde u^n_j)$ in the proof of Lemma (ref), we have \begin{equation} W^{\prime n}= \frac{ k_n^2 }{\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n} [ ( \zeta^{n,\rm v}_{j})^2+ ( \zeta^{n,\rm v}_{j-\mathds{1}/2})^2 ] [ ( \zeta^{n,\rm v}_{j-1})^2+ ( \zeta^{n,\rm v}_{j-1-\mathds{1}/2})^2 ]+ o_p(1). \end{equation} Analogously to the decomposition of $\widetilde V^n$ at the beginning of the proof of Lemma (ref), we can split the last line (without the $o_p(1)$-term) into four terms, say, $W^{\prime n}_1,\dots,W^{\prime n}_4$, each of which is asymptotically equivalent to the ${\cal F}_\infty$-conditional variance of the corresponding $\widetilde V^n_i$-term. To see this, consider $W^{\prime n}_1$ for example, which is given by $ W^{\prime n}_1= \frac{ k_n^2 }{\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n} ( \zeta^{n,\rm v}_{j-1} \zeta^{n,\rm v}_{j})^2$. As usual, by a martingale argument, we can take $\overline\mathbb{E}^n_{[2j-4]^m_n}$-conditional expectation, which turns the previous line exactly into (ref). This shows that $W^{\prime n}_1$ converges in probability to $Q^{(1)}$. Repeating this argument for $W^{\prime n}_2$, $W^{\prime n}_3$ and $W^{\prime n}_4$ completes the proof of the lemma. \end{proof} \begin{proof}[Proof of Lemma (ref)] Throughout this proof, we use $O_p(a_n)$ to signify a term whose $L^2$-norm is bounded by $Ca_n$, where $C$ is a constant independent of ${\omega}$, $i$ and $j$. The notations $o_p(a_n)$ and $o'_p(a_n)$ are used similarly. Also, recall that $H_\rho=0$ by convention if $\varpi>\frac12+\frac12 H$. Our analysis of $\widetilde L^{n,\mathrm{v}}_{2j-\ell}(U)$, $\ell=0,1$, in the proof of Lemma (ref) was independent of whether we are under (ref) or (ref). In particular, we still have (ref). Regarding $\Delta^n_{2j}\widetilde L^{\mathrm{b}}(\widetilde u^n_j)$, we argue as in (ref) and use (ref) and ((ref)) to obtain \begin{align*} \mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_i y}-e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1} y } ]& =\mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1}x}(e^{i(\widetilde u^n_j)_n\Delta^n_i {\epsilon}}-e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1} {\epsilon}})] \\ &\quad+\mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1}{\epsilon}}(e^{i(\widetilde u^n_j)_n\Delta^n_i x}-e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1} x})] \\ &\quad+o'_p(\Delta_n^{H}). \end{align*} Because $\mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1}x}(e^{i(\widetilde u^n_j)_n\Delta^n_i {\epsilon}}-e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1} {\epsilon}})]=O_p(\Delta_n^{H_\rho+(2\varpi-1)_+})$ (cf.\ (ref)) and $\mathbb{E}^n_{i-1}[e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1}{\epsilon}}(e^{i(\widetilde u^n_j)_n\Delta^n_i x}-e^{i(\widetilde u^n_j)_n\Delta^n_{i,i-1} x})]=O_p(\Delta_n^{H+(1-2\varpi)_+})$, we conclude that $ \Delta^n_{2j}\widetilde L^{\mathrm{b}}(\widetilde u^n_j) =O_p(\Delta_n^{H_\ast})$ with $H_\ast = (H+(1-2\varpi)_+)\wedge (H_\rho+(2\varpi-1)_+)$. By the Lévy--Khintchine formula, the mean-value theorem and (ref) and ((ref)), we further have that \begin{align*} &\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j) =\frac{1}{k_n}\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \Bigl\{ e^{-\frac12(\widetilde u^n_j)^2_n c_{(i-1)\Delta_n}\Delta_n}\Psi((\widetilde u^n_j)_n\Delta_n^\varpi\rho_{(i-1)\Delta_n})\\ &\qquad -e^{-\frac12(\widetilde u^n_j)^2_n c_{(i-p_n-1)\Delta_n}\Delta_n }\Psi((\widetilde u^n_j)_n\Delta_n^\varpi\rho_{(i-p_n-1)\Delta_n}) \Bigr\}+o'_p((p_n\Delta_n)^{H}\Delta_n^{(1-2\varpi)_+}). \end{align*} Since the last display is $O_p(\pi_n\theta_n^2)$ by (ref), $1/\sqrt{k_n}=o'((p_n\Delta_n)^H)=o'(\pi_n)$ by ((ref)), $\Delta_n^{H_\ast}=o'(\pi_n)$ and $\log \lvert \widetilde L^{n,\rm s}_{2j-1}(\widetilde u^n_j)\rvert^{-1} \geq C\theta_n^2$ by (ref), we conclude that \[ \Delta^n_{2j} \widetilde {\frak c}(\widetilde u^n_j)= \Re \biggl\{\frac{\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)}{{\frak L}(\widetilde L^n_{2j-1}(\widetilde u^n_j))}\biggr\} + o'_p(\pi_n)= \Re \biggl\{\frac{\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)}{{\frak L}({\cal L}^n_j(\widetilde u^n_j))}\biggr\} + o'_p(\pi_n),\] where the second step follows from $\widetilde L^n_{2j-1}(\widetilde u^n_j)={\cal L}^n_j(\widetilde u^n_j)+o'_p(1)$ (recall the definition after ((ref))). Then $V^{n,\rm{alt}}=V^{n,\rm{alt}}_1+V^{n,\rm{alt}}_2-V^{n,\rm{alt}}_3-V^{n,\rm{alt}}_4+o_p(1)$, where \begin{align*} V^{n,\rm{alt}}_1&=\frac{1}{ {\lvert {\cal J}_n\rvert}\pi_n^2}\sum_{j\in{\cal J}_n}\zeta^{n,\rm s}_{j-1} \zeta^{n,\rm s}_{j}, & V^{n,\rm{alt}}_2& =\frac{1}{ {\lvert {\cal J}_n\rvert}\pi_n^2}\sum_{j\in{\cal J}_n}\zeta^{n,\rm s}_{j-1-\mathds{1}/2} \zeta^{n,\rm s}_{j-\mathds{1}/2}, \\ V^{n,\rm{alt}}_3&= \frac{1}{ {\lvert {\cal J}_n\rvert}\pi_n^2}\sum_{j\in{\cal J}_n}\zeta^{n,\rm s}_{j-1-\mathds{1}/2} \zeta^{n,\rm s}_{j},& V^{n,\rm{alt}}_4&= \frac{1}{ {\lvert {\cal J}_n\rvert}\pi_n^2}\sum_{j\in{\cal J}_n}\zeta^{n,\rm s}_{j-\mathds{1}/2} \zeta^{n,\rm s}_{j-1} \end{align*} and $ \zeta^{n,\rm s}_{j}=\Re \{ {\Delta^n_{2j} \widetilde L^{\mathrm{s}}(\widetilde u^n_j)}/{{\frak L}({\cal L}^n_j(\widetilde u^n_j))} \} $. Next, with the notation $p^n_i=i-p_n-1$, a first-order expansion shows that $\Delta^n_{2j}\widetilde L^{\mathrm{s}}(\widetilde u^n_j)$ is given by\footnote{The second term is set to zero if $\varpi> \frac12+\frac12 H$.} \begin{align*} &-\frac{(\widetilde u^n_j)^2\Delta_n^{(1-2\varpi)_+}}{2k_n}\sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} e^{-\frac12 (\widetilde u^n_j)^2_nc_{p^n_i\Delta_n}\Delta_n}\Psi((\widetilde u^n_j)_n\Delta_n^\varpi \rho_{p^n_i\Delta_n})f'(v_{p^n_i\Delta_n})\\ &\qquad \times\int_0^{(i-1)\Delta_n} [g ((i-1)\Delta_n-s)-g( p^n_i\Delta_n-s )] ({\sigma}^v_sdW_s+\overline{\sigma}^v_sd\overline W_s)\\ &+ \frac{\widetilde u^n_j\Delta_n^{( \varpi-1/2)_+}}{k_n}\sum_{i=(2j-1)p_n+1}^{(2j-1)p_n+k_n} e^{-\frac12 (\widetilde u^n_j)^2_nc_{p^n_i\Delta_n}\Delta_n}\Psi'((\widetilde u^n_j)_n\Delta_n^\varpi \rho_{p^n_i\Delta_n})F'(w_{p^n_i\Delta_n})\\ &\qquad\times\int_0^{(i-1)\Delta_n} [G((i-1)\Delta_n-s)-G( p^n_i\Delta_n-s )] ({\sigma}^w_sdW_s+\overline{\sigma}^w_sd\overline W_s+\widehat{\sigma}^w_sd\widehat W_s) \end{align*} plus an $o'_p(\pi_n)$-error. Note that we have $ \Psi((\widetilde u^n_j)_n\Delta_n^\varpi \rho_{p^n_i\Delta_n})=1+ O(\Delta_n^{(2\varpi-1)_+}\theta_n^2)$ and $ \Psi'((\widetilde u^n_j)_n\Delta_n^\varpi \rho_{p^n_i\Delta_n})=-\widetilde u^n_j \Delta_n^{(\varpi-1/2)_+} \rho_{p^n_i\Delta_n} \mathbb{E}[(\Delta\chi_1)^2]+ O(\Delta_n^{(2\varpi-1)_+}\theta_n^2)$. Besides, given an increasing integer sequence ${\lambda}_n$ such that ${\lambda}_n^{-1}=o'(1)$, we have \begin{align*} &\mathbb{E}\Biggl[\biggl(\int_0^{((i-1)\Delta_n-{\lambda}_n\Delta_n)_+} [g(s-r)-g((i-1)\Delta_n-r)]({\sigma}^v_rdW_r+\overline{\sigma}^v_rd\overline W_r)\biggr)^2\Biggr] \\ &\quad \leq 2K^2K_H^{-2} \int_0^{((i-1)\Delta_n-{\lambda}_n\Delta_n)_+} [(s-r)^{H-1/2}-((i-1)\Delta_n-r)^{H-1/2}]^2 dr\\ &\quad \leq 2K^2K_H^{-2} \int_{{\lambda}_n\Delta_n}^\infty [(r+\Delta_n)^{H-1/2}-r^{H-1/2}]^2 dr\\ &\quad=2K^2K_H^{-2} \Delta_n^{2H}\int_{{\lambda}_n}^\infty [(u+1)^{H-1/2}-u^{H-1/2}]^2 du=o'(\Delta_n^{2H}) \end{align*} for all $s\in((i-1)\Delta_n,i\Delta_n]$, since $u\mapsto[(u+1)^{H-1/2}-u^{H-1/2}]^2$ is integrable. The same type of estimate applies to the integral involving $G$. Together with (ref) (which continues to hold under the alternative hypothesis), it follows that $ \zeta^{n,\rm s}_{j}={\Lambda}^n_{2j} +o_p(\pi_n)$, where $ {\Lambda}^n_j = {\Lambda}^{n,1}_j \mathbf 1_{\{{\Lambda}>0\}}+ {\Lambda}^{n,2}_j \mathbf 1_{\{1-{\Lambda}>0\}}$ and \begin{align*} {\Lambda}^{n,1}_j &=\frac{\Delta_n^{(1-2\varpi)_+}f'(v_{(j-{\lambda}_n-1)p_n\Delta_n})}{k_n \bigl(c_{(j-{\lambda}_n-1)p_n\Delta_n}\mathbf 1_{\{\varpi\geq\frac12\}}+\rho^2_{(j-{\lambda}_n-1)p_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^2]\mathbf 1_{\{\varpi\leq\frac12\}}\bigr)}\\ &\qquad \times\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \int_{(i-{\lambda}_np_n-1)\Delta_n}^{(i-1)\Delta_n} [g ((i-1)\Delta_n-s)\\ &\qquad \qquad\qquad-g( (i-p_n-1)\Delta_n-s )] ({\sigma}^v_{(j-{\lambda}_n-1)p_n\Delta_n}dW_s+\overline{\sigma}^v_{(j-{\lambda}_n-1)p_n\Delta_n}d\overline W_s),\\ {\Lambda}^{n,2}_j &=\frac{2\Delta_n^{(2\varpi-1)_+}\mathbb{E}[(\Delta\chi_1)^2]\rho_{(j-{\lambda}_n-1)p_n\Delta_n} F'(w_{(j-{\lambda}_n-1)p_n\Delta_n})}{k_n \bigl(c_{(j-{\lambda}_n-1)p_n\Delta_n}\mathbf 1_{\{\varpi\geq\frac12\}}+\rho^2_{(j-{\lambda}_n-1)p_n\Delta_n}\mathbb{E}[(\Delta\chi_1)^2]\mathbf 1_{\{\varpi\leq\frac12\}}\bigr)}\\ &\qquad \times\sum_{i=(j-1)p_n+1}^{(j-1)p_n+k_n} \int_{(i-{\lambda}_np_n-1)\Delta_n}^{(i-1)\Delta_n} [G ((i-1)\Delta_n-s)-G( (i-p_n-1)\Delta_n-s )] \\ &\qquad \qquad\qquad\times ({\sigma}^w_{(j-{\lambda}_n-1)p_n\Delta_n}dW_s+\overline{\sigma}^w_{(j-{\lambda}_n-1)p_n\Delta_n}d\overline W_s+\widehat{\sigma}^w_{(j-{\lambda}_n-1)p_n\Delta_n}d\widehat W_s). \end{align*} This shows that $ V^{n,\rm{alt}}_1 =M^n_1+A^n_1+o_p(1)$, where $ M^n_1=\frac{1}{ {\lvert {\cal J}_n\rvert}\pi_n^2}\sum_{j\in{\cal J}_n} \{{\Lambda}^n_{2j} {\Lambda}^n_{2j-2} -\mathbb{E}[{\Lambda}^n_{2j} {\Lambda}^n_{2j-2} \mid {\cal F}_{(2j-{\lambda}_n-3)p_n\Delta_n}] \}$ and $ A^n_1 =\frac{1}{ {\lvert {\cal J}_n\rvert}\pi_n^2}\sum_{j\in{\cal J}_n}\mathbb{E}[{\Lambda}^n_{2j} {\Lambda}^n_{2j-2} \mid {\cal F}_{(2j-{\lambda}_n-3)p_n\Delta_n}]$. The $j$th term in $M^n_1$ is ${\cal F}_{2jp_n\Delta_n}$-measurable with a zero ${\cal F}_{(2j-3-{\lambda}_n)p_n\Delta_n}$-conditional expectation by construction. Therefore, if $j$ and $j'$ are at least $(3+{\lambda}_n)/2$ apart, the two corresponding terms are uncorrelated. Thus, a second moment analysis shows that $M^n_1$ is negligible. On the other hand, $A^n_1$ converges: after shifting the coefficients $c$, $\rho$, $v$, ${\sigma}^v$, $\overline {\sigma}^v$, $\rho$, $w$, ${\sigma}^w$, $\overline{\sigma}^w$ and $\widehat {\sigma}^w$ that appear in ${\Lambda}^n_{2j}$ from $(2j-{\lambda}_n-1)p_n\Delta_n$ to $(2j-{\lambda}_n-3)p_n\Delta_n$, the resulting term is conditionally Gaussian given ${\cal F}_{(2j-{\lambda}_n-3)p_n\Delta_n}$. Therefore, a tedious but entirely straightforward calculation shows that $A^n_1\stackrel{\mathbb{P}}{\longrightarrow} \frac1{T}(C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}}) \int_{I_T} A(t)dt$, where $A(t)$ is defined after ((ref)) and \begin{align*} C_{\kappa,H}&= K_H^{-2} \int_{0}^{1}\int_0^1\int_0^\infty[(\tfrac{v-w}{\kappa}+r+2)^{H-1/2}-(\tfrac{v-w}{\kappa}+r+1)^{H-1/2}]\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times[r^{H-1/2}-(r-1)_+^{H-1/2}] drdwdv. \end{align*} Similarly, $V^{n,\rm{alt}}_2\stackrel{\mathbb{P}}{\longrightarrow} \frac1{T} (C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}})\int_{I_T} A(t-1)dt$. And because $\zeta^{n,\rm s}_{j-\ell_1}$ and $\zeta^{n,\rm s}_{j-\ell_2-\mathds{1}/2}$ are asymptotically uncorrelated for $\ell_1,\ell_2\in\{0,1\}$, we have $V^{n,\rm{alt}}_3+V^{n,\rm{alt}}_4\stackrel{\mathbb{P}}{\longrightarrow}0$, which completes the proof of the first convergence in ((ref)). To see that $C_{\kappa,H}<0$, we change variables from $r$ to $1-s$ to get \begin{align*} C_{\kappa,H} &=K_H^{-2}\int_0^1\int_0^1\int_\mathbb{R} [(\tfrac{v-w}{\kappa}+3-s)_+^{H-1/2}-(\tfrac{v-w}\kappa +2-s)^{H-1/2}_+]\\ &\qquad\qquad\qquad\qquad\qquad\qquad\times[(1-s)^{H-1/2}_+-(-s)_+^{H-1/2}] ds dwdv \\ &=\int_0^1\int_0^1 \mathbb{E}[(B^H_{\frac{v-w}{\kappa}+3}-B^H_{\frac{v-w}{\kappa}+2})(B^H_1-B^H_0)] dvdw, \end{align*} where $B^H$ is a standard fractional Brownian motion with Hurst parameter $H$ (see Mishura08 for the last equation). Since $\lvert (v-w)/\kappa\rvert \leq 1$, the expectation above is the covariance of two non-overlapping increments of fractional Brownian motion with a Hurst index $H<\frac12$, and such a covariance is known to be negative. This shows $C_{\kappa,H}<0$. Finally, let us determine the limit of $W^{n,\rm{alt}}$ in ((ref)). Similarly to the proof of Lemma (ref) (cf.\ (ref)), we have $W^{n,\rm{alt}}=W^{\prime n,\rm{alt}}+W^{\prime\prime n,\rm{alt}}+o_p(1)$, where \begin{align*} W^{\prime n,\rm{alt}}&=\frac{1 }{\pi_n^4\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n} [ ( \zeta^{n,\rm s}_{j})^2+ ( \zeta^{n,\rm s}_{j-\mathds{1}/2})^2 ] [ ( \zeta^{n,\rm s}_{j-1})^2+ ( \zeta^{n,\rm s}_{j-1-\mathds{1}/2})^2 ]\\ W^{\prime\prime n,\rm{alt}}&=\frac{2 }{\pi_n^4\lvert{\cal J}_n\rvert }\sum_{j\in{\cal J}_n} \Bigl\{ 2 \zeta^{n,\rm s}_{j} \zeta^{n,\rm s}_{j-1} \zeta^{n,\rm s}_{j-\mathds{1}/2} \zeta^{n,\rm s}_{j-1-\mathds{1}/2} \\ &\quad - [( \zeta^{n,\rm s}_{j})^2+( \zeta^{n,\rm s}_{j-\mathds{1}/2})^2 ] \zeta^{n,\rm s}_{j-1} \zeta^{n,\rm s}_{j-1-\mathds{1}/2}- [ ( \zeta^{n,\rm s}_{j-1})^2+ ( \zeta^{n,\rm s}_{j-1-\mathds{1}/2})^2 ] \zeta^{n,\rm s}_{j} \zeta^{n,\rm s}_{j-\mathds{1}/2}\Bigr\}. \end{align*} Note that $(\zeta^{n,\rm s}_{j-1},\zeta^{n,\rm s}_{j})$ and $(\zeta^{n,\rm s}_{j-1-\mathds{1}/2},\zeta^{n,\rm s}_{j-\mathds{1}/2})$ are asymptotically uncorrelated. Together with the fact that $\mathbb{E}[X^2Y^2]=\mathbb{E}[X^2]\mathbb{E}[Y^2]+2\mathbb{E}[X_1X_2]^2$ and $\mathbb{E}[X^2Y]=0$ for a centered bivariate Gaussian random vector $(X,Y)$, we deduce, similarly to the analysis of $A^n_1$ above, that $W^{\prime n,\rm{alt}} \stackrel{\mathbb{P}}{\longrightarrow} \frac{2}{T}\int_{I_T} A(t)A(t-1)dt+ \frac{1}{T}(\overline C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+\overline C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}})\int_{I_T} (B(t)+B(t-1))dt$, where $B(t)$ is defined after ((ref)) and $\overline C_{\kappa,H}=2(C_{\kappa,H})^2+(C'_{\kappa,H})^2$ with \begin{align*} C'_{\kappa,H}&=K_H^{-2}\int_0^1\int_0^1\int_0^\infty [(\tfrac{ v-w }{\kappa}+r)^{H-1/2}-(\tfrac{ v-w }{\kappa}+r-1)^{H-1/2}]\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times [r^{H-1/2}-(r-1)_+^{H-1/2}] drdwdv \end{align*} and $W^{\prime\prime n,\rm{alt}} \stackrel{\mathbb{P}}{\longrightarrow} \frac{4}{T}(C_{\kappa,H}\mathbf 1_{\{{\Lambda}\in\{\frac12,1\}\}}+C_{\kappa,H_\rho}\mathbf 1_{\{{\Lambda}=0\}})^2\int_{I_T} A(t)A(t-1) dt$. \end{proof}