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 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.