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

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  • Zakhar Kabluchko

    (Ulm University)

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.

Suggested Citation

  • Zakhar Kabluchko, 2012. "Limit Laws for Sums of Independent Random Products: the Lattice Case," Journal of Theoretical Probability, Springer, vol. 25(2), pages 424-437, June.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:2:d:10.1007_s10959-010-0296-5
    DOI: 10.1007/s10959-010-0296-5
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    References listed on IDEAS

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    1. Durrett, Richard, 1979. "Maxima of branching random walks vs. independent random walks," Stochastic Processes and their Applications, Elsevier, vol. 9(2), pages 117-135, November.
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