Uniform convergence in some limit theorems for multiple particle systems

For n particles diffusing throughout R (or Rd), let ηn,t(A), A ϵ B, t ⩾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 {(ηn,t(ø) − Eηn,t(ø))√n:t ϵ [0,M], ø ϵ CαL(R)}, n ϵ 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.