Annals of Probability
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 dQn(x) = zn-1 exp[ -Hn(x)]Π dP(xj), where x = (x1,..., xn) ∈ Rn, where Hn is the Hamiltonian, zn is a normalizing constant and P is a probability measure on R. For certain forms of the Hamiltonian Hn, Ellis and Newman (1978b) showed that under appropriate conditions on P, there exists an integer r ≥ 1 such that Sn/n1-1/2r 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 Hn, 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
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.