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 finite-dimensional 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.
"Weak convergence of measure-valued processes and r-point functions." Ann. Probab. 35 (5) 1769 - 1782, September 2007. https://doi.org/10.1214/009117906000001088