Gaussian approximation of the empirical process under random entropy conditions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:119

Article

Settati, Adel

关键词: Empirical processes;    Strong invariance principle;    Random entropy;   

DOI  :  10.1016/j.spa.2008.08.001

来源: Elsevier

【 摘 要 】

We obtain rates of strong approximation of the empirical process indexed by functions by a Brownian bridge under only random entropy conditions. The results of Berthet and Mason [P. Berthet, D.M. Mason, Revisiting two strong approximation results of Dudley and Philipp, in: High Dimensional Probability, in: IMS Lecture Notes-Monograph Series, vol. 51, 2006, pp. 155-172] under bracketing entropy are extended by combining their method to properties of the empirical entropy. Our results show that one can improve the universal rate v(n) = o(root log log n) from Dudley and Philipp [R.M. Dudley, W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrsch. Verw. Gebiete 62 (1983) 509-552] into v(n) -> 0 at a logarithmic rate, under a weak random entropy assumption which is close to necessary. As an application the results of Kolchinskii [V.I. Kolchinskii, Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions, J. Theoret. Probab. 7 (1994) 73-118] are revisited when the conditions coming in addition to random entropy are relaxed. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

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