Quantitative Breuer-Major Theorems

Ivan Nourdin (), Giovanni Peccati () and Mark Podolskij ()
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Ivan Nourdin: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Postal: Boîte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France
Giovanni Peccati: University of Luxembourg, Postal: Rue Richard, Coudenhove-Kalergi, L-1359 Luxembourg

CREATES Research Papers from Department of Economics and Business Economics, Aarhus University

Abstract: We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n f(X_k)$, $n\geq 1$, 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 $\E[f^2(X_1)]$ 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.

Keywords: Berry-Esseen bounds; Breuer-Major central limit theorems; Gaussian processes; Interpolation; Malliavin calculus; Stein’s method (search for similar items in EconPapers)
JEL-codes: C10 C60 (search for similar items in EconPapers)
Pages: 24
Date: 2010-05-12
New Economics Papers: this item is included in nep-ecm and nep-ore
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Related works:
Journal Article: Quantitative Breuer-Major theorems (2011)
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Persistent link: https://EconPapers.repec.org/RePEc:aah:create:2010-22

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