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

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

2007-10-16

arxiv.org/abs/0710.2998v1

Annals of Probability 2007, Vol. 35, No. 5, 1769-1782

Mathematics

Probability

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

Scientific paper

10.1214/009117906000001088

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.

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