LARGE DEVIATION LOCAL LIMIT-THEOREMS FOR RATIO STATISTICS

Abstract

Let {T(n), n greater-than-or-equal-to 1} be an arbitrary sequence of nonlattice random variables and let {S(n), n greater-than-or-equal-to 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic R(n) = T(n)/S(n) based on simple conditions on the moment generating functions of T(n) and S(n). When S(n) = n, our main result reduces to that of Chaganty and Sethuraman[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when T(n) and S(n) are both lattice random variables. We call our theorems large deviation local limit theorems for R(n), since the conditions of our theorems imply that R(n) --> c in probability for some constant c. We present some examples to illustrate our theorems.