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Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. (English) Zbl 1189.60084

Summary: 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 \(V_{n}=\sum _{0\leq k\leq n-1} H_{q}(B_{k+1}-B_{k})\). 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 of \(Z_{n}\) and of \(Z\), 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 I. Nourdin and G. Peccati [Probab. Theory Relat. Fields 145, No. 1–2, 75–118 (2009; Zbl 1175.60053)] in the case where \(H<1-1/(2q)\), corresponding to the case where one has normal approximation.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60F05 Central limit and other weak theorems
60H07 Stochastic calculus of variations and the Malliavin calculus

Citations:

Zbl 1175.60053
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