Limit theorems for sums of dependent random variables occurring in statistical mechanics

Summary We study the asymptotic behavior of partial sums S_nfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number , and a positive integer k so that (S_n–nm)/n^1–1/2k converges weakly to a random variable with density proportional to exp(–¦s¦^2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.Alfred P. Sloan Research Fellow. Research supported in part by a Broadened Faculty Research Grant at the University of Massachusetts and by National Science Foundation Grant MPS 76-06644Research supported in part by National Science Foundation Grants MPS 74-04870 A01 and MCS 77-20683