Total variation bounds for Gaussian functionals

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Pratelli, Luca¹  Rigo, Pietro² 

[1] Accademia Navale, Viale Italia 72, I-57100 Livorno, Italy

[2] Univ Pavia, Dipartimento Matemat F Casorati, Via Ferrata 1, I-27100 Pavia, Italy

关键词: Fractional brownian motion;    Gaussian process;    Stable convergence;    Total variation distance;   

DOI  :  10.1016/j.spa.2018.07.005

来源: Elsevier

【 摘 要 】

Let X = {X-t : 0 <= t <= 1} be a centered Gaussian process with continuous paths, and I-n = a(n)/2 integral(1 )(0)t(n-1) (X-1(2)- X-1(2)) dt where the a(n) are suitable constants. Fix beta is an element of (0, 1), c(n) > 0 and c > 0 and denote by N-c the centered Gaussian kernel with (random) variance cX(1)(2). Under a Holder condition on the covariance function of X, there is a constant k(beta) such that parallel to P(root c(n) I-n is an element of.) - E[N-c(.)] parallel to <= k(beta) (a(n)/n(1+alpha))(beta) + vertical bar c(n) - c vertical bar/c for all n >= 1, where parallel to .parallel to is total variation distance and alpha the Holder exponent of the covariance function. Moreover, if a(n)/n(1+alpha) -> 0 and c(n) -> c, then root c(n )I(n) converges parallel to .parallel to-stably to N-c, in the sense that parallel to P-F(root c(n) I-n is an element of.) - E-F [N-c (.)] parallel to -> 0 for every measurable F with P(F) > 0. In particular, such results apply to X = fractional Brownian motion. In that case, they strictly improve the existing results in Nourdin et al. (2016) and provide an essentially optimal rate of convergence. (C) 2018 Elsevier B.V. All rights reserved.

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