Weak convergence of measure-valued processes and $r$-point functions

Abstract

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.

Published in at http://dx.doi.org/10.1214/009117906000001088 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (4)

• Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions
• On the range of lattice models in high dimensions - extended version
• An extension of the inductive approach to the lace expansion
• The survival probability and r-point functions in high dimensions