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 where {Xj}j=1[infinity] is a sequence of i.i.d.r.v.s with values in the measurable space , is a measurable function for each t[set membership, variant]T, {an} is an arbitrary sequence of real numbers and cn(t) is a real number, for each t[set membership, variant]T and each n[greater-or-equal, slanted]1. We also consider the weak convergence of processes of the form where {Xj}j=1[infinity] is a sequence of independent r.v.s with values in the measurable space , and is a measurable function for each t[set membership, variant]T. Instead of measuring the size of the brackets using the strong or weak Lp norm, we use a distance inherent to the process. We present applications to the weak convergence of stochastic processes satisfying certain Lipschitz conditions.