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| Title: | Limit laws for sums of random exponentials
Research supported in part by DFG Grant 436 RUS 113/534 and NSF Grant DMS-9971592. |
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| DOI No: | 10.1142/9789812702241_0004 | |
| Source: | RECENT DEVELOPMENTS IN STOCHASTIC ANALYSIS AND RELATED TOPICS (pp 45-65) | |
| Author(s): | Gérard Ben Arous
Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Leonid Bogachev Department of Statistics, University of Leeds, Leeds LS2 9JT, United Kingdom Stanislav Molchanov Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA |
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| Abstract: | We study the limiting distribution of the sum  as t → ∞, N → ∞, where (Xi) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida’s Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = − log P{Xi > x} is regularly varying at infinity with index 1 < ϱ < ∞. An appropriate scale for the growth of N relative to t is of the form eλH0(t), where the rate function H0(t) is a certain asymptotic version of the cumulant generating function H(t) = log E[etXi] provided by Kasahara’s exponential Tauberian theorem. We have found two critical points, 0< λ1< λ2 < ∞, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below λ2, we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent α = α(ϱ, λ) ranging from 0 to 2 and with skewness parameter β = 1. A limit theorem for the maximal value of the sample {etXi, i = 1, ..., N} is also proved. |
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| Keywords: | Random exponentials; sums of independent random variables; regular variation; Kasahara’s Tauberian theorem; weak limit theorems; stable laws AMSC numbers: Primary 60G50, Primary 60F05, Secondary 60E07 |
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| Full Text: | View full text in PDF format (739KB) | |
| TOC: | Back to Table of Contents | |
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