Electronic Communications in Probability

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

Jean-Christophe Breton and Ivan Nourdin

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Let $q\geq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $H\in$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the $q$th Hermite polynomial. For any $n\geq 1$, set $V_n=\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. 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 $\mathscr{L}(Z_n)$ and $\mathscr{L}(Z)$, 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 Nourdin and Peccati (2007) in the case where $H<1-1/(2q)$, corresponding to the case where one has normal approximation.

Article information

Electron. Commun. Probab., Volume 13 (2008), paper no. 46, 482-493.

Accepted: 26 September 2008
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus

Total variation distance Non-central limit theorem Fractional Brownian motion Hermite power variation Multiple stochastic integrals Hermite random variable

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Breton, Jean-Christophe; Nourdin, Ivan. Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electron. Commun. Probab. 13 (2008), paper no. 46, 482--493. doi:10.1214/ECP.v13-1415. https://projecteuclid.org/euclid.ecp/1465233473

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