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Quantitative Breuer-Major theorems

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 nor...

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Hauptverfasser: Nourdin, Ivan (VerfasserIn) , Peccati, Giovanni (VerfasserIn) , Podolskij, Mark (VerfasserIn)
Dokumenttyp: Article (Journal)
Sprache:Englisch
Veröffentlicht: April 2011
In: Stochastic processes and their applications
Year: 2011, Jahrgang: 121, Heft: 4, Pages: 793-812
ISSN:1879-209X
DOI:10.1016/j.spa.2010.12.006
Online-Zugang:Verlag, Volltext: http://dx.doi.org/10.1016/j.spa.2010.12.006
Verlag, Volltext: http://www.sciencedirect.com/science/article/pii/S0304414910002917
Volltext
Verfasserangaben:Ivan Nourdin, Giovanni Peccati, Mark Podolskij

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520 |a 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. 
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