# Quantitative Breuer-Major Theorems

## Author Info

• 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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## Bibliographic Info

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

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Length: 24
Date of revision:
Handle: RePEc:aah:create:2010-22

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Web page: http://www.econ.au.dk/afn/

## Related research

Keywords: Berry-Esseen bounds; Breuer-Major central limit theorems; Gaussian processes; Interpolation; Malliavin calculus; Stein’s method;

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Find related papers by JEL classification:
• C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
• C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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## References

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1. Corinne Berzin & José León, 2007. "Estimating the Hurst Parameter," Statistical Inference for Stochastic Processes, Springer, vol. 10(1), pages 49-73, 01.
2. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Multipower Variation for Brownian Semistationary Processes," CREATES Research Papers 2009-21, School of Economics and Management, University of Aarhus.
3. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2007. "Power variation for Gaussian processes with stationary increments," CREATES Research Papers 2007-42, School of Economics and Management, University of Aarhus.
4. 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.
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## Citations

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Cited by:
1. Pham, Viet-Hung, 2013. "On the rate of convergence for central limit theorems of sojourn times of Gaussian fields," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2158-2174.

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