WEAK CONVERGENCE OF MEASURE-VALUED PROCESSES AND r-POINT FUNCTIONS

We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finitedimensional distributions. The conditions are convergence of the Fourier transform of the r-point functions and perhaps convergence of the “survival probabilities.” These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.