Exponential inequalities and functional central limit theorems for random fields

LSTA, Université de Paris 6, 175 rue du Chevaleret, 75013 Paris Cedex 05, France; dedecker@ccr.jussieu.fr.

Received: 9 April 1999
Revised: 7 June 2001

Abstract

We establish new exponential inequalities for partial sums of random fields. Next, using classical chaining arguments, we give sufficient conditions for partial sum processes indexed by large classes of sets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, the condition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.