# Quantitative Breuer-Major Theorems

Abstract : We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ 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 $|\E[h(S_n)] -\E[h(S)]|$, 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 $Var[f^2(X_1)]$ and on simple infinite series involving the components of $r$. In particular, our results generalize and refine some classic CLTs by Breuer-Major, Giraitis-Surgailis and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time-series.
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Journal articles
Stochastic Processes and their Applications, Elsevier, 2011
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Cited literature [39 references]

https://hal.archives-ouvertes.fr/hal-00484096
Contributor : Ivan Nourdin <>
Submitted on : Friday, June 4, 2010 - 2:00:41 PM
Last modification on : Thursday, April 27, 2017 - 9:46:38 AM
Document(s) archivé(s) le : Thursday, September 23, 2010 - 12:57:28 PM

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### Identifiers

• HAL Id : hal-00484096, version 2
• ARXIV : 1005.3107

### Citation

Ivan Nourdin, Giovanni Peccati, Mark Podolskij. Quantitative Breuer-Major Theorems. Stochastic Processes and their Applications, Elsevier, 2011. 〈hal-00484096v2〉

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