A limit theorem for symmetric statistics of Brownian particles

Asymptotic distributions for a family of time-varying symmetric statistics formed from an infinite particle system are derived and a representation for the limit is obtained in terms of multiple stochastic integrals. This family arises from a system of Brownian particles diffusing in R whose initial configuration is given via a Poisson point process on R. It is shown that a symmetric statistic of order p in this family can be considered as an element of C{[0,T],S′(Rp)} and as the rate of the Poisson process approaches infinity these symmetric statistics converge in distribution as random elements of the above mentioned function space. A stochastic partial differential equation satisfied by the limit is obtained. Finally, a representation for the limit as a mixed multiple stochastic integral with respect to a space-time white noise and a white noise on R, is derived.