Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion

TL;DR

This paper establishes error bounds for the non-normal approximation of Hermite power variations of fractional Brownian motion when the Hurst index exceeds a critical value, extending previous normal approximation results.

Contribution

It derives upper bounds for the total variation distance in the non-normal case, filling a gap in the understanding of Hermite power variations for certain Hurst indices.

Findings

Provides explicit bounds for the total variation distance

Extends previous results to non-normal regimes

Clarifies the behavior of Hermite variations for H > 1 - 1/(2q)

Abstract

Let $q\geq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $H\in(0,1)$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the Hermite polynomial having degree $q$. For any $n\geq 1$, set $V_n=\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper is to derive, in the case when the Hurst index verifies $H>1-1/(2q)$, an upper bound for the total variation distance between the laws $\mathscr{L}(Z_n)$ and $\mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization of $V_n$ which converges in distribution towards $Z$. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case when $H<1-1/(2q)$, corresponding to the situation where one has normal approximation.