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

Authors

Narasinga Rao Chaganty, Old Dominion UniversityFollow
Jayaram Sethuraman

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 dP(x_j), where x = (x₁,..., x_n) ∈ Rⁿ, 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