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 λ, and a positive integer k so that (S n−nm)/n1−1/2k converges weakly to a random variable with density proportional to exp(−λ¦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.
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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 76-06644
Research supported in part by National Science Foundation Grants MPS 74-04870 A01 and MCS 77-20683
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Ellis, R.S., Newman, C.M. Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrscheinlichkeitstheorie verw Gebiete 44, 117–139 (1978). https://doi.org/10.1007/BF00533049
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DOI: https://doi.org/10.1007/BF00533049