Quantitative Breuer-Major theorems

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Nourdin, Ivan²  Peccati, Giovanni³  Podolskij, Mark¹ 

[1] Swiss Fed Inst Technol, Dept Math, CH-8092 Zurich, Switzerland

[2] Univ Paris 06, Lab Probabil & Modeles Aleatoires, F-75252 Paris 5, France

[3] Fac Sci Technol & Commun, UR Math, L-1359 Luxembourg, Luxembourg

关键词: Berry-Esseen bounds;    Breuer-Major central limit theorems;    Gaussian processes;    Interpolation;    Malliavin calculus;    Stein's method;   

DOI  :  10.1016/j.spa.2010.12.006

来源: Elsevier

【 摘 要 】

We consider sequences of random variables of the type S-n = n(-1/2) Sigma(n)(k=1){f(X-k) - E[f(X-k)]}, n >= 1, where X = (X-k)(k is an element of Z) is a d-dimensional Gaussian process and f : R-d -> R is a measurable function. It is known that, under certain conditions on f and the covariance function r of X, S-n converges in distribution to a normal variable S. In the present paper we derive several explicit upper bounds for quantities of the type vertical bar E[h(S-n)] - E[h(S)]vertical bar, where h is a sufficiently smooth test function. Our methods are based on Malliavin calculus, on interpolation techniques and on the Stein's method for normal approximation. The bounds deduced in our paper depend only on Var[f(X-1)] and on simple infinite series involving the components of r. In particular, our results generalize and refine some classic CLTs given by Breuer and Major, Giraitis and Surgailis, and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time series. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free