Limit Theorems in the Area of Large Deviations for Some Dependent Random Variables

Abstract

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 $(X^{(n)}_1,\ldots, X^{(n)}_n)$. Standard theory of physics would dictate that the joint distribution of the spins can be modelled by $dQ_n(\mathbf{x}) = z^{-1}_n \exp\lbrack -H_n(\mathbf{x})\rbrack\Pi dP(x_j)$, where $\mathbf{x} = (x_1,\ldots, x_n) \in \mathscr{R}^n$, where $H_n$ is the Hamiltonian, $z_n$ is a normalizing constant and $P$ is a probability measure on $\mathscr{R}$. For certain forms of the Hamiltonian $H_n$, Ellis and Newman (1978b) 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 Chaganty and Sethuraman (1985), 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.