Limit Laws for Sums of Independent Random Products: the Lattice Case

Abstract

Let {Vi,j;(i,j)∈ℕ2} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent random products $$Z_n=\sum_{i=1}^{N_n}\prod_{j=1}^{n}e^{V_{i,j}}$$ as n,Nn→∞ have been investigated by a number of authors. Depending on the growth rate of Nn, the random variable Zn obeys a central limit theorem or has limiting α-stable distribution. The latter result is true for non-lattice Vi,j only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence Zn fails to converge in distribution, it is relatively compact in the weak topology, and we describe its cluster set. This set is a topological circle consisting of semi-stable distributions.