A matrix limit theorem with applications to probability theory

The classical Poisson limit theorem studies the limit laws of Sn where Sn=∑ n j=1Xjn and X1n,...,Xnn 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 (An(s))n where s is a real parameter and An(s) is a finite dimensional (the dimension being fixed) matrix of the form An(s)=R(s)+n−1(Q(s)+Bn(s)) where lim n→∞ Bn(s)=0. This problem seem to be of independent interest but does not appear to have been treated in the literature