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2008 Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion
Jean-Christophe Breton, Ivan Nourdin
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Electron. Commun. Probab. 13: 482-493 (2008). DOI: 10.1214/ECP.v13-1415

Abstract

Let $q\geq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $H\in$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the $q$th Hermite polynomial. 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 where $H<1-1/(2q)$, corresponding to the case where one has normal approximation.

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Jean-Christophe Breton. Ivan Nourdin. "Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion." Electron. Commun. Probab. 13 482 - 493, 2008. https://doi.org/10.1214/ECP.v13-1415

Information

Accepted: 26 September 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1189.60084
MathSciNet: MR2447835
Digital Object Identifier: 10.1214/ECP.v13-1415

Subjects:
Primary: 60F05
Secondary: 60G15, 60H07

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