## Document Type

Article

## Publication Date

1987

## DOI

10.1214/aop/1176992162

## Publication Title

Annals of Probability

## Volume

15

## Issue

2

## Pages

628-645

## 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,..., 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)]Π *d*P(x_{j}), where x = (x_{1},..., x_{n}) ∈ *R*^{n}, where *H*_{n} is the Hamiltonian, *z _{n}* is a normalizing constant and P is a probability measure on 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 ≥ 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 ≥ 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.

## Original Publication Citation

Chaganty, N. R., & Sethuraman, J. (1987). Limit theorems in the area of large deviations for some dependent random variables. *Annals of Probability, 15*(2), 628-645. doi:10.1214/aop/1176992162

## Repository Citation

Chaganty, Narasinga Rao and Sethuraman, Jayaram, "Limit Theorems in the Area of Large Deviations for Some Dependent Random Variables" (1987). *Mathematics & Statistics Faculty Publications*. 165.

https://digitalcommons.odu.edu/mathstat_fac_pubs/165