Weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions

We study the weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions involving majorizing measures. As an application, we consider the weak convergence of stochastic processes of the form {(a(n)(-1)Sigma f(Xj,t)) - c(n)(t) : t is an element of T}, n greater than or equal to 1, where {X-j)(j=1)(infinity) is a sequence of i.i.d.r.v.s with values in the measurable space (S, Y), f(., t) : S --> R is a measurable function for each t is an element of T, {a(n)} is an arbitrary sequence of real numbers and c(n)(t) is a real number, for each t is an element of T and each n greater than or equal to 1. We also consider the weak convergence of processes of the form {Sigma(j=1)(n) f(j)(X-j,t) : t is an element of T}, n greater than or equal to 1, where {X-j}(j=1)(infinity) is a sequence of independent r.v.s with values in the measurable space (S-j, Y-j), and f(j)(., t) : S-j --> R is a measurable function for each t is an element of T. Instead of measuring the size of the brackets using the strong or weak L-p norm, we use a distance inherent to the process. We present applications to the weak convergence of stochastic processes satisfying certain Lipschitz conditions. (C) 1998 Elsevier Science B.V. All rights reserved.