Bulletin of the
Korean Mathematical Society
BKMS

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

Article

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Bull. Korean Math. Soc. 2014; 51(2): 317-328

Printed March 31, 2014

https://doi.org/10.4134/BKMS.2014.51.2.317

Copyright © The Korean Mathematical Society.

The second central limit theorem for martingale difference arrays

Jongsig Bae, Doobae Jun, and Shlomo Levental

Sungkyunkwan University, Gyeongsang National University, Michigan State University

Abstract

In Bae {\em et al.} \cite{ReferBJL}, we have considered the uniform CLT forthe martingale difference arrays under the uniformly integrable entropy. In this paper, we prove the same problem under the bracketing entropy condition. The proofs are based on Freedman inequality combined with a chaining argument that utilizes majorizing measures. The results of present paper generalize those for a sequence of stationary martingale differences. The results also generalize independent problems.

Keywords: central limit theorem, martingale difference array, bracketing entropy, majorizing measure, eventual uniform equicontinuity

MSC numbers: Primary 60F17; Secondary 60F05