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

Abstract

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 ∑ j=1 n f(X j,t) −c n(t):t∈T , n⩾1, where { X j } j=1 ∞ is a sequence of i.i.d.r.v.s with values in the measurable space (S, S) , f( · ,t):S→ R is a measurable function for each t∈ T, { a n } is an arbitrary sequence of real numbers and c n ( t) is a real number, for each t∈ T and each n⩾1. We also consider the weak convergence of processes of the form ∑ j=1 n f j(X j,t):t∈T , n⩾1, where { X j } j=1 ∞ is a sequence of independent r.v.s with values in the measurable space (S j, S j) , and f j( · ,t):S j→ R is a measurable function for each t∈ 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.