Limit theorems in the area of large deviations for some dependent random variables
A magnetic body can be considered to consist of n sites, where n is large. The magnetic spins at these n sites, whose sum is the total magnetization present in the body, can be modelled by a triangular array of random variables \((X_ 1^{(n)},...,X_ n^{(n)})\). Standard theory of physics would dictate that the joint distribution of the spins can be modelled by \[ dQ_ n(x)=z_ n^{-1} \exp [-H_ n(x)]\prod dP(x_ j), \] where \(x=(x_ 1,...,x_ n)\in {\mathcal R}^ n\), where \(H_ n\) is the Hamiltonian, \(z_ n\) is a normalizing constant and P is a probability measure on \({\mathcal R}.\) For certain forms of the Hamiltonian \(H_ n\), \textit{R. S. Ellis} and \textit{C. M. Newman} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 117-139 (1978; Zbl 0364.60120)] showed that under appropriate conditions on P, there exists an integer \(r\geq 1\) such that \(S_ n/n^{1-1/2r}\) converges in distribution to a random variable. This limiting random variable is Gaussian if \(r=1\) and non-Gaussian if \(r\geq 2.\) In this article, utilizing the large deviation local limit theorems for arbitrary sequences of random variables of \textit{N. R. Chaganty} and \textit{J. Sethuraman} [Ann. Probab. 13, 97-114 (1985; Zbl 0559.60030)], we obtain similar limit theorems for a wider class of Hamiltonians \(H_ n\), which are functions of moment generating functions of suitable random variables. We also present a number of examples to illustrate our theorems.
scientific article; zbMATH DE number 4038920