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  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  and other results of independent problems. The results also generalizes those of Bae and Choi  to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with $n$ is given.