Rate of Convergence for the Weighted Hermite Variations of the Fractional Brownian Motion

Abstract

In this paper, we obtain a rate of convergence in the central limit theorem for high order weighted Hermite variations of the fractional Brownian motion. The proof is based on the techniques of Malliavin calculus and the quantitative stable limit theorems proved by Nourdin et al. (Ann Probab 44:1–41, 2016).

Appendix

Here we prove a result we need in the proof of Theorem 1.1. Recall the notation $D_{k/n} F = \langle DF, \delta _{k/n}\rangle _{\mathfrak {H}}$ where $\delta _{k/n} =\mathbb {1}_{[k/n,(k+1)/n]}$.

Lemma 4.1

For any integers $\ell , M, a,b,c$ such that $M\ge 0$, $0\le c\le q$, $0\le a\le q-c$, $0\le b\le c$ and any real number $p>1$,there exists a constant C depending on q, M, p, and the Hurst parameter H such that

$$\begin{aligned} \left\| \underbrace{D_{k/n}\cdots D_{k/n}}_{a\,\mathrm{times}} \left( u_n \otimes _{b} \delta _{k/n}^{\otimes c} \right) \right\| _{M,p} \le C\Vert f\Vert _{M+a,p}n^{\kappa (a,b,c,H)}, \end{aligned}$$

where

$$\begin{aligned}&\kappa (a,b,c,H) :={\left\{ \begin{array}{ll} {-1/2-H(2a+c)} &{} b>0, H\le 1/2\\ -1/2-a-H(c-b)+ (-Hb)\\ \vee (q(H-1)+1-b) &{} b>0, H>1/2\\ {-a(2H\wedge 1)-Hc} &{} b=0 \end{array}\right. }. \end{aligned}$$

Proof

For $0\le m\le M$, let

$$\begin{aligned} A_m&:=E\left[ \left\| D^m\left( \underbrace{D_{k/n}\cdots D_{k/n}}_{a\,\mathrm{times}} \left( u_n \otimes _{b} \delta _{k/n}^{\otimes c} \right) \right) \right\| _{\mathfrak {H}^{\otimes (m+q-2b+c)}}^p\right] ^{1/p}\\&=n^{qH-1/2}E\left[ \left\| \sum _{j=0}^{n-1} f^{(m+a)}(B_{j/n}) \alpha _{k,j/n}^{a} \beta _{j,k}^b\delta _{k/n}^{\otimes (c-b)} \delta _{j/n}^{\otimes (q-b)} {\otimes } \varepsilon _{j/n}^{\otimes m} \right\| _{\mathfrak {H}^{\otimes (m+q-2b+c)}}^p\right] ^{1/p}. \end{aligned}$$

Taking the norm in $\mathfrak {H}^{m+q-2b+c}$, we have

$$\begin{aligned} A_m\le & {} Cn^{qH-1/2} \Vert f\Vert _{m+a,p}\\&\left\{ \sum _{j,j'=0}^{n-1} |\alpha _{k,j/n}|^a |\alpha _{k,j'/n}|^a |\beta _{j,k}|^b| \beta _{j',k}|^b n^{-2H(c-b)} |\beta _{j,j'}|^{q-b} \right\} ^{1/2}. \end{aligned}$$

We consider three different cases:

Case 1 Suppose that $0<b<q$. Applying Lemma 2.4 to $\sum _{j,j'=0}^{n-1}|\beta _{j,k}|^b|\beta _{j',k}|^b |\beta _{j,j'}|^{q-b}$ and Lemma 2.1(a) to each of the $\alpha$ terms, we have

$$\begin{aligned} A_m&\le C n^{qH-1/2} \Vert f\Vert _{m+a,p} \left( n^{-2a(2H\wedge 1)}n^{-2H(c-b)} n^{(-2H(q+b))\vee (2-2(q+b))}\right) ^{1/2}\\&=C n^{qH-1/2} \Vert f\Vert _{m+a,p} n^{-a(2H\wedge 1)} n^{-H(c-b)} n^{(-H(q+b))\vee (1-(q+b))}. \end{aligned}$$

When $H\le 1/2$, $q+b\ge 2\ge 1/(1-H)$, so $-H(q+b) \ge 1-(q+b)$, and we have

$$\begin{aligned} A_m \le C n^{qH-1/2} \Vert f\Vert _{m+a,p} n^{-H(q+2a+c)}. \end{aligned}$$

When $H>1/2$,

$$\begin{aligned} A_m&\le Cn^{qH-1/2} \Vert f\Vert _{m+a,p}n^{-a} n^{-H(c-b)} n^{(-H(q+b))\vee (1-(q+b))} \\&=Cn^{qH-1/2} \Vert f\Vert _{m+a,p}n^{-a-H(c-b)}n^{(-H(q+b))\vee (1-(q+b))}\\&=C \Vert f\Vert _{m+a,p}n^{-a-1/2-H(c-b)}n^{(-Hb)\vee (q(H-1)+1-b)}. \end{aligned}$$

Case 2 Suppose that $b=q$. In this case, $c=q$ and $a=0$ and, applying Lemma 2.2(a),

$$\begin{aligned} A_m&\le C n^{qH-1/2} \Vert f\Vert _{m,p} \left[ \sum _{j,j'=0}^{n-1} |\beta _{j,k}|^q|\beta _{j',k}|^q\right] ^{1/2}\\&= Cn^{qH-1/2} \Vert f\Vert _{m,p} \sum _{j=0}^{n-1} |\beta _{j,k}|^q\\&\le C \Vert f\Vert _{m,p}n^{qH-1/2} n^{(-2qH)\vee (1-2q)}. \end{aligned}$$

Note that $-2qH = (-2qH)\vee (1-2q)$. Thus, this estimate coincides with the estimate in case 1 when $b=c=q$ and $a=0$.

Case 3 Suppose that $b=0$. Applying Lemma 2.2(b) to $\sum _{j,j'=0}^{n-1} |\beta _{j,j'}|^q$ and Lemma 2.1(a) to each of the $\alpha$ terms,

$$\begin{aligned} A_m&\le C n^{qH-1/2} \Vert f\Vert _{m+a,p} n^{-a(2H\wedge 1)} n^{-Hc} n^{(1/2-qH)\vee (1-q)}\\&= C n^{qH-1/2} \Vert f\Vert _{m+a,p}n^{-a(2H\wedge 1)}n^{-Hc} n^{1/2-qH} \\&\le C \Vert f\Vert _{m+a,p}n^{-a(2H\wedge 1)-Hc}. \end{aligned}$$

This concludes the proof of the lemma. $\square$