Limit laws for sums of independent random products: the lattice case

TL;DR

This paper investigates the asymptotic behavior of sums of independent random products in the lattice case, revealing that the normalized sums form a cluster set of semi-stable distributions rather than converging.

Contribution

It extends the limit law analysis to the lattice case, showing the normalized sums are relatively compact and characterizing their cluster set as a circle of semi-stable distributions.

Findings

Normalized sums are relatively compact in the weak topology.

Cluster set forms a topological circle of semi-stable distributions.

Convergence in distribution fails in the lattice case.

Abstract

Let $\{V_{i,j}; (i,j)\in\N^2\}$ be a two-dimensional array of i.i.d.\ 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 $n,N_n\to\infty$ have been investigated by a number of authors. Depending on the growth rate of $N_n$, the random variable $Z_n$ obeys a central limit theorem, or has limiting $\alpha$-stable distribution. The latter result is true for non-lattice $V_{i,j}$ only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence $Z_n$ fails to converge in distribution, it is relatively compact in the weak topology, and describe its cluster set. This set is a topological circle consisting of semi-stable distributions.