Error bounds on the non-normal approximation of hermite power variations of fractional brownian motion

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 qth Hermite polynomial. For any n ≥1, set Vn = Σk=0n−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 ℒ(Zn) and ℒ(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. © 2008 Applied Probability Trust.