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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 46 open journal systems 


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

Jean-Christophe Breton, Laboratoire Mathématiques, Image et Applications
Ivan Nourdin, Laboratoire de Probabilités et Modèles Aléatoires


Abstract
Let q≥2 be a positive integer, B be a fractional Brownian motion with Hurst index H∈(0,1), Z be an Hermite random variable of index q, and Hq denote the q-th Hermite polynomial. For any n≥1, set Vn=∑0≤k≤n-1 Hq(Bk+1-Bk). 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 of Zn and of Z, where Zn stands for the correct renormalization of Vn 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.


Full text: PDF

Pages: 482-493

Published on: September 26, 2008
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Electronic Communications in Probability. ISSN: 1083-589X