Strong approximation of the empirical distribution function for absolutely regular sequences in R^d.

Abstract : We prove a strong approximation result with rates for the empirical process associated to an absolutely regular stationary sequence of random variables with values in R^d. As soon as the absolute regular coefficients of the sequence decrease more rapidly than n^{1-p} for some p in ]2,3], we show that the error of approximation between the empirical process and a two-parameter Gaussian process is of order n^{1/p} (\log n)^c(d) for some positive c(d) depending on d, both in L^1 and almost surely. The power of n being independent of the dimension, our results are even new in the independent setting, and improve earlier results. In addition, for absolutely regular sequences, we show that the rate of approximation is optimal up to the logarithmic term.
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Article dans une revue
Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (no. 9, 56 pp.)
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https://hal.archives-ouvertes.fr/hal-00798305
Contributeur : Emmanuel Rio <>
Soumis le : vendredi 8 mars 2013 - 12:14:31
Dernière modification le : jeudi 9 février 2017 - 15:55:52

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Jérôme Dedecker, Florence Merlevède, Emmanuel Rio. Strong approximation of the empirical distribution function for absolutely regular sequences in R^d.. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (no. 9, 56 pp.). <hal-00798305>

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