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

By J. Breton, Ivan Nourdin. 2008

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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 $V_n=\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. 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.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c9f6e880b097b5adbbe66d0f4b72555ec908fc2d

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Citations in the corpus

Variations and estimators for selfsimilarity parameters via Malliavin calculus (2009, background): “We refer to [19] for the one-dimensional case and H ∈ (0, 3 4), [ 2 ] for then one-dimensional case and H ∈[ 3 4,1) and to [20] for the multidimensional case and H ∈ (0, 3 4).”
Optimal rates for parameter estimation of stationary Gaussian processes (2016, methodology): “These results’ proofs, which are omitted, follow the results in Remark 7 and from the tools in this section and those in [20] and [5].”
Variations and Hurst index estimation for a Rosenblatt process using longer filters (2009, background): “Recent interest in the Rosenblatt and other Hermite processes, due in part to their non-Gaussian character, and in part for their independent mathematical value, is evidenced by the following references: [4, 6, 10, 18, 19, 20, 27, 28].”
Quantitative Breuer-Major Theorems (2010, background): “More to the point, slight variations in the form of f and r may imply that either the normalization by the factor n 1 / 2 is inappropriate, or the limiting distribution is not-Gaussian (or both): see Dobrushin and Major [16], Rosenblatt [33] and Taqqu [38, 39] for several classic results connected t”
Stein’s method on Wiener chaos (2007, background): “39 pages; Two sections added; To appear in PTRF. 2008. <hal-00199024v5> ha l- 00 19 90 24 , v er si on 5 - 1 0 M ay 2 00 8 Stein's method on Wiener haosby Ivan Nourdin∗ and Giovanni Pe ati†University of Paris VIRevised version: May 10, 2008Abstra t: We ombine Malliavin al ulus with Stein's method, i”
ASYMPTOTIC THEORY FOR ESTIMATING DRIFT PARAMETERS IN THE FRACTIONAL VASICEK MODEL (2018, methodology): “When κ > 0, Kleptsyna and Le Breton (2002), Tudor and Viens (2007), Tanaka (2014) studied the maximum likelihood (ML) estimator; Hu and Nualart (2010) studied the least squares (LS) estimator; Tanaka (2013) studied the minimum contrast (MC) estimator; Hu and Nualart (2010) introduced and studied an ”