Some remarks on the uniform weak convergence of stochastic processes

Abstract

It is known that the uniform weak convergence of a sequence of stochastic processes is equivalent to the finite dimensional convergence, plus an asymptotic equicontinuity condition with respect to any pseudometric which makes the parameter space totally bounded. Here, we see that in this characterization, it is possible to take an intrinsic metric of the process. This is applied to obtain necessary and sufficient conditions for the weak convergence of local U-processes.