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Weak convergence to derivatives of fractional Brownian motion
The $p$-dimensional fractionally integrated process of Type II (e.g., Marinucci and Robinson, 1999), is given by
This expression defines the operator $\Delta_{+}^{-d}=(1-L)_{+}^{-d}$ as a finite sum, and the fractional coefficients $\pi_{n}(d)$ are defined by the binomial expansion of $(1-z)^{-d}$. That is, \[ \pi_{n}(d)=(-1)^{n}\binom{-d}{n}=d(d+1)\dots(d+n-1)/n!\sim cn^{d-1} \] with \textquotedblleft$\sim$\textquotedblright\ denoting that the ratio of the left- and right-hand sides converges to one. The parameter $d$ is called the memory parameter, which we assume satisfies $d>1/2$. Throughout, $\xi_{t}$ is a $p$-dimensional linear process,
for some $p\times p$ coefficient matrices $C_{j}$ and a $p$-dimensional innovation sequence, $\varepsilon_{t}$, which is independently and identically distributed (i.i.d.)\ with mean zero and variance matrix $\Sigma$ (precise conditions will be given in Section (ref)).
We define the normalized process $Z_{\left\lfloor Tr\right\rfloor }(d)=T^{1/2-d}\Delta_{+}^{-d}\xi_{\left\lfloor Tr\right\rfloor }$ for $d>1/2$ and $r\in\lbrack0,1]$, where $\left\lfloor \cdot\right\rfloor $ denotes the integer-part of the argument. The functional central limit theorem\footnote{Even earlier results were available for the so-called Type I process; e.g.\ Davydov (1970) and Taqqu (1975).} (FCLT) for $Z_{\left\lfloor Tr\right\rfloor }(d)$ was proved by Akonom and Gourieroux (1987) for ARMA processes $\xi_{t}$, and by Marinucci and Robinson (2000) for linear processes $\xi_{t}$ with coefficients satisfying a summability condition; see Assumption (ref) below. In particular, these authors showed that
where $\Gamma(\cdot)$ is the Gamma function, $W$ is Brownian motion with variance matrix $C(1)\Sigma C(1)^{\prime}$, $C(1)=\sum_{j=-\infty}^{\infty }C_{j}$, and \textquotedblleft$\Rightarrow$\textquotedblright\ denotes weak convergence in the space of c\`{a}dl\`{a}g functions on $[0,1]$ endowed with the Skorokhod topology; see Billingsley (1968) for a general treatment. That is, the normalized process $Z_{[Tr]}(d)$ converges weakly to fractional Brownian motion (fBm), $W(r;d)$, which is also of Type II; see Marinucci and Robinson (1999) for a detailed comparison of Types I and II fBm.
In fact, the results in Marinucci and Robinson (2000) also imply weak convergence of the derivative of $\Delta_{+}^{-d}\xi_{t}$, suitably normalized. We use $\mathsf{D}_{d}^{m}$ to denote the $m$'th order derivative with respect to $d$. Differentiating term-by-term we find $\mathsf{D}_{d} \pi_{n}(d)=\pi_{n}(d)\sum_{k=0}^{n-1}(k+d)^{-1}$; see Appendix A of Johansen and Nielsen (2016) for additional details on the fractional coefficients and their derivatives. With this notation, Marinucci and Robinson (2000) proved that
Thus, because of the factor $\sum_{k=0}^{n-1}(k+d)^{-1}\sim\log n$, a different normalization is needed, but the weak limit is still fBm.
Related to ((ref)) and ((ref)), Hualde (2012) showed the limit result\footnote{There is a missing minus sign in either (6) or (8) in Hualde (2012). Of course, this is irrelevant for the marginal distribution of $A(r;d)$ because $A(r;d)$ is a zero-mean Gaussian process. However, the sign is critical when considering the joint distribution of $A(r;d)$ and $W(r;d)$, for example.}
where $A(r;d)=\Gamma(d)^{-1}\int_{0}^{r}\log(r-s)(r-s)^{d-1}\mathsf{d}W(s)$ was denoted a \textquotedblleft modified fBm\textquotedblright. The derivation of ((ref)) was motivated by a regression analysis of so-called \textquotedblleft unbalanced cointegration\textquotedblright, where the process $A(r;d)$ enters in the asymptotic distribution theory; see Hualde (2012, 2014). Note, however, that $A(r;d)=\Gamma^{-1}(d)\mathsf{D}_{d} (\Gamma(d)W(r;d))$ is not the derivative of fBm.
In this paper, we prove related results for weak convergence of the derivatives with respect to $d$ of $Z_{\left\lfloor Tr\right\rfloor }(d)$ to corresponding derivatives of fBm. Differentiating term-by-term as in ((ref)) we find
In the general case, the coefficients in the linear representation of $\mathsf{D}_{d}^{m}Z_{t}(d)$ will be calculated by recursion; see Section (ref)\ and Lemma (ref). Note the relation
In recent work, Johansen and Nielsen (2021) generalize earlier work on statistical inference in the fractionally cointegrated vector autoregressive model (Johansen and Nielsen, 2012b) to allow each variable in the multivariate process to have its own fractional parameter (integration order). They call this the \textquotedblleft multifractional\textquotedblright\ vector autoregressive model. One interpretation of this model is a generalization of Hualde's (2014) bivariate unbalanced cointegrated regression model to a multivariate system framework. Johansen and Nielsen (2021) show that, in this setting, the derivative $\mathsf{D}_{d}Z_{\left\lfloor Tr\right\rfloor }(d)$ and its weak limit $\mathsf{D}_{d}W(r;d)$ play an important role in the asymptotic distribution theory for the maximum likelihood estimators of the fractional parameters. We present some details of this analysis in Section (ref) to motivate and apply our results.
In Section (ref) we show that the result ((ref)) of Hualde (2012) can be generalized to allow for weights $(-\sum_{k=n}^{T}(k+d)^{-1})^{m}$ for any integer $m\geq0$. In Section (ref) we use this result together with ((ref)) of Marinucci and Robinson (2000) to show weak convergence of $\mathsf{D}_{d}^{m}Z_{\left\lfloor Tr\right\rfloor }(d)$ to derivatives of fBm. The application of our results to the multifractional cointegration model is given in Section (ref), and some concluding remarks are given in Section (ref). In the next section, however, we first consider $m=1$, because the arguments simplify substantially in that case.
In this section, we apply the results of Marinucci and Robinson (2000) in ((ref)) and Hualde (2012) in ((ref)) to show that the first derivative of the fractional process, i.e.$\ \mathsf{D}_{d}Z_{\left\lfloor Tr\right\rfloor }(d)$, converges weakly to $\mathsf{D}_{d}W(r;d)$. Precise conditions under which the results hold will be stated in Section (ref) before we give the general results.
The derivative $\mathsf{D}_{d}Z_{\left\lfloor Tr\right\rfloor }(d)$ is rewritten, using ((ref)) and $\sum_{k=0}^{n-1}(k+d)^{-1}=\sum_{k=0} ^{T}(k+d)^{-1}-\sum_{k=n}^{T}(k+d)^{-1}$, as
Here, $Z_{\left\lfloor Tr\right\rfloor }(d)\Rightarrow W(r;d)$ and $H_{\left\lfloor Tr\right\rfloor }(d)\Rightarrow A(r;d)$ by ((ref)) and ((ref)), respectively. Strictly speaking, ((ref)) and ((ref)) need to hold jointly, but that is a consequence of Theorem (ref) below.
To evaluate the factor $-\log T+\sum_{k=0}^{T}(k+d)^{-1}$ in ((ref)), recall the following definition and series expansion of the Digamma function, \[ \psi(d)=\mathsf{D}_{d}\log\Gamma(d)=-\gamma-\sum_{k=0}^{\infty}((k+d)^{-1} -(k+1)^{-1})\text{ for }d\neq0,-1,\dots, \] where $\gamma=\lim_{T\rightarrow\infty}(\sum_{k=1}^{T}k^{-1}-\log T)=0.577\dots$ is the Euler-Mascheroni constant; see Abramowitz and Stegun (1972, eqns.\ 6.3.1 and 6.3.16). We then find that
Finally we prove that
By definition of the derivative, \[ \mathsf{D}_{d}\int_{0}^{r}(r-s)^{d-1}\mathsf{d}W(s)-\int_{0}^{r} \log(r-s)(r-s)^{d-1}\mathsf{d}W(s)=\lim_{\delta\rightarrow0}K_{d}(\delta), \] where \[ K_{d}(\delta)=\delta^{-1}\int_{0}^{r}(r-s)^{d-1}((r-s)^{\delta}-1-\delta \log(r-s))\mathsf{d}W(s). \] By the mean value theorem, \[ (r-s)^{\delta}-1-\delta\log(r-s)=\frac{1}{2}\delta^{2}\log^{2} (r-s)(r-s)^{\delta^{\ast}}\text{ for }|\delta^{\ast}|\leq|\delta|. \] Hence we find, using the Frobenius norm $\left\Vert A\right\Vert =(\operatorname*{tr}\{A^{\prime}A\})^{1/2}$,
because $2d-1>0$. This proves ((ref)).
Combining these results, it follows that
Thus, the first derivative of the fractional process $Z_{\left\lfloor Tr\right\rfloor }(d)$ converges weakly to the first derivative of the fBm $W(r;d)$. Interestingly, the above arguments leading to ((ref)) required only the weak convergences in ((ref)) and ((ref)) (jointly) together with some well-known results regarding the Digamma function. Consequently, our result ((ref)) holds whenever ((ref)) and ((ref)) hold jointly. In the next two sections we will prove the corresponding result for derivatives of any order under precisely stated conditions.
In this section, we generalize the result ((ref)) of Hualde (2012). To this end, we define the processes
so that $Z_{\left\lfloor Tr\right\rfloor }(d)=H_{0,\left\lfloor Tr\right\rfloor }(d)$ and $H_{\left\lfloor Tr\right\rfloor } (d)=H_{1,\left\lfloor Tr\right\rfloor }(d)$. In Theorem (ref) below we find the joint weak limit of $H_{m,\left\lfloor Tr\right\rfloor }(d)$, $m=0,1,\dots,M$, for any non-negative integer $M$, but first we state our assumptions.
We note that the moment condition in Assumption (ref) is in fact necessary; see Johansen and Nielsen (2012a). The rank conditions in Assumptions (ref)--(ref) ensure that the long-run variance of $\xi_{t}$ is positive definite.
Assumptions (ref)--(ref) are identical to the corresponding conditions in Hualde (2012) and Marinucci and Robinson (2000). Thus, ((ref)), ((ref)), and the results in Section (ref), and in particular the weak convergence in ((ref)), all hold under Assumptions (ref)--(ref).
We next analyze the derivatives of the fractional process $Z_{\left\lfloor Tr\right\rfloor }(d)$ with respect to the fractional parameter $d$, i.e.\ $\mathsf{D}_{d}^{m}Z_{\left\lfloor Tr\right\rfloor }(d)$. In terms of the fractional coefficients and their derivatives, $\mathsf{D}_{d} ^{m}Z_{\left\lfloor Tr\right\rfloor }(d)$ can be defined recursively as follows. We apply logarithmic differentiation and let
where the coefficients $R_{Tn}^{(m)}(d)$ are defined by the relation $\mathsf{D}_{d}^{m}(T^{1/2-d}\pi_{n}(d))=T^{1/2-d}\pi_{n}(d)R_{Tn}^{(m)}(d)$. We note that \[ \mathsf{D}_{d}^{m+1}Z_{\left\lfloor Tr\right\rfloor }(d)=\sum_{n=0} ^{\left\lfloor Tr\right\rfloor -1}T^{1/2-d}\pi_{n}(d)(\mathsf{D}_{d} R_{Tn}^{(m)}(d)+R_{Tn}^{(1)}(d)R_{Tn}^{(m)}(d))\xi_{\left\lfloor Tr\right\rfloor -n}, \] so that the coefficients $R_{Tn}^{(m)}(d)$ must satisfy the recursion
To illustrate the recursion, the next two terms of $R_{Tn}^{(m)}(d)$ are
There is a similar recursive definition of the derivatives of fBm. We define $R^{(m)}(d)$ by the relation $\mathsf{D}_{d}^{m}(\Gamma(d)^{-1}(r-s)^{d-1} )=\Gamma(d)^{-1}(r-s)^{d-1}R^{(m)}(d)$ and find
The first equality in ((ref)) follows by the same proof as for ((ref)). As in ((ref)) and ((ref)) we find that the functions $R^{(m)}(d)$ must satisfy the recursion
To compare with $R_{Tn}^{(2)}(d)$ and $R_{Tn}^{(3)}(d)$, we find
where $\psi^{(j)}(d)=\mathsf{D}_{d}^{j}\psi(d)=\mathsf{D}_{d}^{j+1}\log \Gamma(d)$ denotes the polygamma function; see Abramowitz and Stegun (1972, eqn.\ 6.4.1). The recursive formulations in ((ref)) and ((ref)) are clearly much more tractable than direct calculation for larger values of $m$. We note, in particular, the strong similarity between the terms $R_{Tn}^{(m)}(d)$ and $R^{(m)}(d)$. For example, for $m=1$ and with $n$ replaced by $\left\lfloor Tr\right\rfloor -\left\lfloor Ts\right\rfloor $, we find that \[ R_{T,\left\lfloor Tr\right\rfloor -\left\lfloor Ts\right\rfloor } ^{(1)}(d)=-\log T+\sum_{k=0}^{T}(k+d)^{-1}-\sum_{k=\left\lfloor Tr\right\rfloor -\left\lfloor Ts\right\rfloor }^{T}(k+d)^{-1}\rightarrow -\psi(d)+\log(r-s)=R^{(1)}(d) \] as $T\rightarrow\infty$; c.f.\ ((ref)).
We next derive the solutions to the recursions.
We are now ready to give our main result.
One motivation for the results on the weak convergence of derivatives of the fractional process comes from the analysis of the multifractional cointegrated vector autoregressive (MFCVAR) model; see Johansen and Nielsen (2021). Let $d=(d_{1},\dots,d_{p})^{\prime}$ be a vector of fractional parameters and let $b$ be a scalar fractional parameter. The MFCVAR model with parameters $\lambda=(d,b,\alpha,\beta,\Omega)$ and no lags is given by
where the matrix differencing operator is $\Lambda_{+}(d)=\operatorname*{diag} (\Delta_{+}^{d_{1}},\dots,\Delta_{+}^{d_{p}})$ and $\varepsilon_{t}$ satisfies Assumption (ref). In particular, $\varepsilon_{t}$ is i.i.d.\ with mean zero and variance $\Omega$.
The properties of the solution to these equations can be found from the corresponding result for the FCVAR model studied in Johansen and Nielsen (2012b). We denote true values by subscript zero, and in particular $d_{0p}$ denotes the $p$'th element of $d_{0}$. Now, if we define $\tilde{X}_{t}$ by $\Delta_{+}^{d_{p}}\tilde{X}_{t}=\Lambda_{+}(d)X_{t}$, then $\tilde{X}_{t}$ is given by the equations
These equations define the FCVAR model of Johansen and Nielsen (2012b) with scalar fractional parameters $d_{p}$ and $b$ together with $(\alpha ,\beta,\Omega)$. It follows from Theorem 2 of Johansen and Nielsen (2012b) that the solution to ((ref)), for $(d_{p},b,\alpha,\beta,\Omega )=(d_{0p},b_{0},\alpha_{0},\beta_{0},\Omega_{0})$, is \[ \tilde{X}_{t}=C_{0}\Delta_{+}^{-d_{p_{0}}}\varepsilon_{t}+\Delta_{+} ^{b_{0}-d_{0p}}Y_{t}, \] where $C_{0}=\beta_{0\bot}(\alpha_{0\bot}^{\prime}\beta_{0\bot})^{-1} \alpha_{0\bot}^{\prime}$ and $Y_{t}$ is a stationary linear process satisfying Assumption (ref). Consequently, the solution to ((ref)), for $\lambda=\lambda_{0}$, satisfies
This shows that $\Delta_{+}^{d_{i0}}X_{it}$ is in general fractional of order zero, for $i=1,\dots,p$, so that the model ((ref)) allows each component of $X_{t}$ to have its own fractional order, and is therefore called \textquotedblleft multifractional\textquotedblright. Pre-multiplying ((ref)) by $\beta_{0}^{\prime}$ shows that $\Delta_{+}^{-b_{0}} \beta_{0}^{\prime}\Lambda_{+}(d_{0})X_{t}$ is also fractional of order zero; that is, some linear combinations of the processes $\{\Delta_{+}^{d_{i0} -b_{0}}X_{it}\}_{i=1}^{p}$ are fractional of order zero and hence $X_{t}$ is cointegrated.
We define the i.i.d.\ process $\xi_{t}=(\alpha_{0\bot}^{\prime}\beta_{0\bot })^{-1}\alpha_{0\bot}^{\prime}\varepsilon_{t}$ such that $C_{0}\varepsilon _{t}=\beta_{0\bot}\xi_{t}$. The three processes $Z_{t}(b_{0})$, $Z_{t}^{\ast }(b_{0})$, and $\mathsf{D}_{b_{0}}Z_{t}(b_{0})$ are then defined in terms of $\xi_{t}$ as in ((ref)), ((ref)), and ((ref)), respectively. It follows from the above analysis that $Z_{\left\lfloor Tr\right\rfloor } (b_{0})$ and $Z_{\left\lfloor Tr\right\rfloor }^{\ast}(b_{0})$ converge weakly to fractional Brownian motion $W(r;b_{0})$ and that $\mathsf{D}_{b_{0} }Z_{\left\lfloor Tr\right\rfloor }(b_{0})$ converges weakly to $\mathsf{D} _{b_{0}}W(r;b_{0})$, and that the processes converge jointly.
To simplify the subsequent analysis we assume that $\Omega=\Omega_{0}$, $d_{p}=d_{0p}$, $\alpha=\alpha_{0}$, and $b=b_{0}>1/2$. This allows us to focus on the parameters that give rise to \textquotedblleft non-standard\textquotedblright\ asymptotic distributions, and in particular to the application of $\mathsf{D}_{b_{0}}W(r;b_{0})$. Specifically, we define the parameters $\theta=\beta_{0\bot}^{\prime}\beta$\ (or $\beta=\beta_{0} +\bar{\beta}_{0\bot}\theta$ with $\bar{A}=A(A^{\prime}A)^{-1}$ for any matrix $A$ with full rank) and $\gamma_{i}=d_{i}-d_{i0}$ for $i=1,\dots,p$, such that $\gamma_{p}=0$. With this notation we can define the residual, using ((ref)) and ((ref)), as \[ \varepsilon_{t}(\theta,\gamma)=(I_{p}-\alpha_{0}(\beta_{0}^{\prime} +\theta^{\prime}\bar{\beta}_{0\bot}^{\prime})(1-\Delta_{+}^{-b_{0}} ))\Lambda_{+}(\gamma)(C_{0}\varepsilon_{t}+\Delta_{+}^{b_{0}}Y_{t}), \] and the Gaussian likelihood is \[ L_{T}(\theta,\gamma)=-\frac{1}{2}\operatorname*{tr}\{\Omega_{0}^{-1}T^{-1} \sum_{t=1}^{T}\varepsilon_{t}(\theta,\gamma)\varepsilon_{t}(\theta ,\gamma)^{\prime}\}=-\frac{1}{2}\operatorname*{tr}\{\Omega_{0}^{-1} M_{T}(\varepsilon(\theta,\gamma),\varepsilon(\theta,\gamma))\}, \] where $M_{T}(a,b)=T^{-1}\sum_{t=1}^{T}a_{t}b_{t}^{\prime}$. We will use this simple model to illustrate the role of the processes $Z_{t}(b_{0})$ and $\mathsf{D}_{b_{0}}Z_{t}(b_{0})$ and their limits in the analysis of the score functions for $\gamma$ and $\theta$ evaluated at $\lambda_{0}$.
The derivative of $\varepsilon_{t}(\theta,\gamma)$ with respect to $\theta$ at $\lambda=\lambda_{0}$ in the direction $\partial\theta\in\mathbb{R} ^{(p-r)\times r}$ is denoted $\mathsf{D}_{\theta}\varepsilon_{t} |_{\lambda=\lambda_{0}}(\partial\theta)$ and similarly for $\mathsf{D} _{\gamma}\varepsilon_{t}|_{\lambda=\lambda_{0}}(\partial\gamma),\partial \gamma\in\mathbb{R}^{p}$, but with $\partial\gamma_{p}=0$ because $\gamma _{p}=0$. We find
where $Z_{t}$ and $Z_{t}^{\ast}$ are given in ((ref)) and ((ref)), and where we use `$\simeq$' to indicate that equality holds up to a stationary process that disappears asymptotically when we normalize the nonstationary processes. We identify the score vector $S_{T,\theta}$ for $\theta$ from $\mathsf{D}_{\theta}L_{T}|_{\lambda=\lambda_{0}}(\partial\theta )=(\operatorname{vec}\partial\theta)^{\prime}S_{T,\theta}$, and similarly for $\gamma$. We then find that
and, using $\operatorname*{tr}\{A^{\prime}B\}=(\operatorname{vec}A)^{\prime }\operatorname{vec}B$, the scores are
Here we have defined the $(p-r)r\times p$ matrix $B_{0}=(\beta_{0}^{\prime }e_{1}\otimes\beta_{0\bot}^{\prime}e_{1},\dots,\beta_{0}^{\prime}e_{p} \otimes\beta_{0\bot}^{\prime}e_{p})$, with $e_{i}$ denoting the $i$'th unit vector in $\mathbb{R}^{p}$, and used the property that $\operatorname*{tr} \{\beta_{0}^{\prime}\operatorname*{diag}(\phi)\beta_{0\bot}M\}=\phi^{\prime }B_{0}^{\prime}\operatorname{vec}M$; see Theorem 2 of Johansen and Nielsen (2021). Thus, $S_{T,\theta}\in\mathbb{R}^{(p-r)r}$ and $S_{T,\gamma} \in\mathbb{R}^{p}$.
We note that the product moments $T^{1/2}M_{T}(Z(b_{0}),\varepsilon)$ and $T^{1/2}M_{T}(Z^{\ast}(b_{0}),\varepsilon)$ converge jointly to their weak limit $\int_{0}^{1}W(r;b_{0})\mathsf{d}W^{\prime}(r)$, so the scores become linearly dependent in the limit. We therefore use the relation ((ref)) to eliminate $Z_{t}^{\ast}(b_{0})=Z_{t}(b_{0})+(\log T)^{-1}\mathsf{D}_{b_{0}}Z_{t}(b_{0})$, and the score for $\gamma$ becomes \[ T^{-b_{0}+1}S_{T,\gamma}\simeq-B_{0}^{\prime}\operatorname{vec}(T^{1/2} M_{T}((\log T)Z(b_{0})+\mathsf{D}_{b_{0}}Z(b_{0}),\varepsilon)\Omega_{0} ^{-1}\alpha_{0}). \] We can now eliminate the linear dependence in the limit by defining the new parameter \[ \operatorname{vec}\tilde{\theta}=\operatorname{vec}\theta+(\log T)B_{0} \gamma\in\mathbb{R}^{(p-r)r} \] and \[ \tilde{\varepsilon}_{t}(\operatorname{vec}\tilde{\theta},\gamma)=\varepsilon _{t}(\operatorname{vec}\theta,\gamma)=\varepsilon_{t}(\operatorname{vec} \tilde{\theta}-(\log T)B_{0}\gamma,\gamma). \] Then the scores and their joint limits become
Thus, the introduction of the derivative of the fractional process and its limit allows one to reparametrize the score to find a mixed Gaussian asymptotic distribution, which can then be exploited to conduct inference for some hypotheses in the MFCVAR model. For a detailed analysis we refer to Johansen and Nielsen (2021).
Weak convergence of derivatives of fractional processes is interesting in its own right. However, it is also likely to find application in statistical analysis of inference problems related to multivariate fractional processes.
Hualde (2012) motivated his result ((ref)) with a bivariate regression analysis of so-called \textquotedblleft unbalanced cointegration\textquotedblright\ (see Hualde, 2014), but also anticipated that results like ((ref)) may be useful in the statistical analysis of polynomial co-fractionality (see Johansen, 2008, and Franchi, 2010).
In Section (ref) we presented an application of our results in ((ref))\ and Theorem (ref) to the asymptotic distribution theory for the maximum likelihood estimators of the fractional parameters in the so-called \textquotedblleft multifractional\textquotedblright\ vector autoregressive model of Johansen and Nielsen (2021). In this setting, the derivative $\mathsf{D}_{d}Z_{\left\lfloor Tr\right\rfloor }(d)$ and its weak limit $\mathsf{D}_{d}W(r;d)$ play an important role because they allow avoiding linear dependence in the limit and because the asymptotic distribution is expressed in terms of both $W(r,d)$ and $\mathsf{D}_{d}W(r,d)$.