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.
Citation
Narasinga Rao Chaganty. Jayaram Sethuraman. "Limit Theorems in the Area of Large Deviations for Some Dependent Random Variables." Ann. Probab. 15 (2) 628 - 645, April, 1987. https://doi.org/10.1214/aop/1176992162
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