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.
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