The uniform CLT for martingale difference arrays under the uniformly integrable entropy
Bull. Korean Math. Soc. 2010 Vol. 47, No. 1, 39-51
https://doi.org/10.4134/BKMS.2010.47.1.39
Printed January 1, 2010
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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