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

Zakhar Kabluchko ()
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Zakhar Kabluchko: Ulm University

Journal of Theoretical Probability, 2012, vol. 25, issue 2, 424-437

Abstract: Abstract Let {V i,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,N n →∞ 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 α-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 we describe its cluster set. This set is a topological circle consisting of semi-stable distributions.

Keywords: Random products; Random exponentials; Semi-stable laws; Random energy model; Triangular arrays; Central limit theorem; 60G50; 60F05; 60F10 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10959-010-0296-5

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