Quantitative Breuer-Major theorems
We consider sequences of random variables of the type , n>=1, where is a d-dimensional Gaussian process and 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 , 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 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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Volume (Year): 121 (2011)
Issue (Month): 4 (April)
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- 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.
- 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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- 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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