A matrix limit theorem with applications to probability theory

Abstract

The classical Poisson limit theorem studies the limit laws of S_n where S_n=∑ ⁿ _j=1X_jn and X_1n,...,X_nn is a sequence of {0, 1} valued, independent, identically distributed random variables. In this paper we will weaken the independence assumption and investigate the possible limit laws for certain types of dependent sequences. This leads us to the study of the limit of (A_n(s))ⁿ where s is a real parameter and A_n(s) is a finite dimensional (the dimension being fixed) matrix of the form A_n(s)=R(s)+n⁻¹(Q(s)+B_n(s)) where lim _n→∞ B_n(s)=0. This problem seem to be of independent interest but does not appear to have been treated in the literature.