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.
Jongsig Bae, Doobae Jun, and Shlomo Levental
Sungkyunkwan University, Sungkyunkwan University, and Michigan State University
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
2014; 51(2): 317-328
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd