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


 
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1. Title Title of document Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion
 
2. Creator Author's name, affiliation, country Jean-Christophe Breton; Laboratoire Mathématiques, Image et Applications
 
2. Creator Author's name, affiliation, country Ivan Nourdin; Laboratoire de Probabilités et Modèles Aléatoires
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Total variation distance; Non-central limit theorem; Fractional Brownian motion; Hermite power variation; Multiple stochastic integrals; Hermite random variable
 
3. Subject Subject classification 60F05; 60G15; 60H07
 
4. Description 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 $q$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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7. Date (YYYY-MM-DD) 2008-09-26
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1415
 
10. Identifier Digital Object Identifier 10.1214/ECP.v13-1415
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 13
 
12. Language English=en
 
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