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Gaussian approximation of the empirical process under random entropy conditions

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  • Settati, Adel

Abstract

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 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 vn-->0 at a logarithmic rate, under a weak random entropy assumption which is close to necessary. As an application the results of Koltchinskii [V.I. Kolchinskii, Komlós-Major-Tusnády 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.

Suggested Citation

  • Settati, Adel, 2009. "Gaussian approximation of the empirical process under random entropy conditions," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1541-1560, May.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:5:p:1541-1560
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    Cited by:

    1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP44/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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