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

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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 Hermite polynomial having degree q. For any n≥ 1, set Vₙ=∑ₖ₌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 when H<1-1/(2q), corresponding to the situation where one has normal approximation.