# Quantitative Breuer-Major Theorems

## Author Info

Listed author(s):
• Ivan Nourdin

()

(Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie)

• Giovanni Peccati

()

(University of Luxembourg)

• Mark Podolskij

()

(ETH Zürich and CREATES)

## 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.

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File URL: ftp://ftp.econ.au.dk/creates/rp/10/rp10_22.pdf

## Bibliographic Info

Paper provided by Department of Economics and Business Economics, Aarhus University in its series CREATES Research Papers with number 2010-22.

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 Length: 24 Date of creation: 12 May 2010 Handle: RePEc:aah:create:2010-22 Contact details of provider: Web page: http://www.econ.au.dk/afn/

## References

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1. León, José & Ludeña, Carenne, 2007. "Limits for weighted p-variations and likewise functionals of fractional diffusions with drift," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 271-296, March.
2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
3. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij & Jeannette H.C. Woerner, 2008. "Bipower variation for Gaussian processes with stationary increments," CREATES Research Papers 2008-21, Department of Economics and Business Economics, Aarhus University.
4. Corinne Berzin & José León, 2007. "Estimating the Hurst Parameter," Statistical Inference for Stochastic Processes, Springer, vol. 10(1), pages 49-73, 01.
5. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Multipower Variation for Brownian Semistationary Processes," CREATES Research Papers 2009-21, Department of Economics and Business Economics, Aarhus University.
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