Quantitative Breuer-Major theorems

• Verfasst von: Nourdin, Ivan [VerfasserIn]
• Verfasst von: Peccati, Giovanni [VerfasserIn]
• Verfasst von: Podolskij, Mark [VerfasserIn]
• Titel: Quantitative Breuer-Major theorems
• Verf.angabe: Ivan Nourdin, Giovanni Peccati, Mark Podolskij
• Umfang: 20 S.
• Fussnoten: Available online: 22 December 2010 ; Gesehen am 29.05.2018
• Titel Quelle: Enthalten in: Stochastic processes and their applications
• Jahr Quelle: 2011
• Band/Heft Quelle: 121(2011), 4, S. 793-812
• ISSN Quelle: 1879-209X
• DOI: doi:10.1016/j.spa.2010.12.006
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1575817314

Abstract

We consider sequences of random variables of the type Sn=n−1/2∑k=1n{f(Xk)−E[f(Xk)]}, n≥1, where X=(Xk)k∈Z is a d-dimensional Gaussian process and f:Rd→R is a measurable function. It is known that, under certain conditions on f and the covariance function r of X, Sn converges in distribution to a normal variable S. In the present paper we derive several explicit upper bounds for quantities of the type |E[h(Sn)]−E[h(S)]|, 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 V ar[f(X1)] 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.