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 an−1∑j=1nf(Xj,t)−cn(t):t∈T, n⩾1,where {Xj}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, {an} is an arbitrary sequence of real numbers and cn(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=1nfj(Xj,t):t∈T, n⩾1,where {Xj}j=1∞ is a sequence of independent r.v.s with values in the measurable space (Sj,Sj), and fj( · ,t):Sj→R is a measurable function for each t∈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.