Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2010; 47(1): 39-51

Printed January 1, 2010

https://doi.org/10.4134/BKMS.2010.47.1.39

Copyright © The Korean Mathematical Society.

The uniform CLT for martingale difference arrays under the uniformly integrable entropy

Jongsig Bae, Doobae Jun, and Shlomo Levental

Sungkyunkwan University, Sungkyunkwan University, and Michigan State University

Abstract

In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with $n$ is given.

Keywords: uniform CLT, martingale difference array, uniformly integrable entropy, restricted chaining, sequential empirical process

MSC numbers: Primary 60F17; Secondary 60F05

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