Uniform convergence in some limit theorems for multiple particle systems

For n particles diffusing throughout R (or Rd), let [eta]n,t(A), A [epsilon] B, t [greater-or-equal, slanted]0, be the random measure that counts the number of particles in A at time t. It is shown that for some basic models (Brownian particles with or without branching and diffusion with a simple interaction) the processes {([eta]n,t(ø) - E[eta]n,t(ø))/[radical sign]n:t [epsilon] [0,M], ø [epsilon] C[alpha]L(R)}, n [epsilon] N, converge in law uniformly in (t, ø). Previous results consider only convergence in law uniform in t but not in ø. The methods used are from empirical process theory.