Ben Arous, G, Bogachev, LV and Molchanov, SA (2004) Limit laws for sums of random exponentials. In: Albeverio, S, Ma, ZM and Roeckner, M, (eds.) Recent Developments in Stochastic Analysis and Related Topics. (1st SinoGerman Conference on Stochastic Analysis, Beijing, China, Aug 29  Sep 03, 2002). World Scientific , 45  65. ISBN 9812561048
Abstract
We study the limiting distribution of the sum SN(t) = Sigma(i=1)(N) e(tXi) as t > infinity, N > infinity, 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 logtail distribution function h(x) = log P{XI > x} is regularly varying at infinity with index 1 < rho < infinity. An appropriate scale for the growth of N relative to t is of the form e(lambda H0(t)), where the rate function Ho(t) is a certain asymptotic version of the cumulant. generating function H(t) = log E[e(tXi)] provided by Kasahara's exponential Tauberian theorem. We have found two critical points, 0 < lambda(1) < lambda(2) < infinity, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below lambda(2), we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent alpha = alpha(rho, lambda) ranging from 0 to 2 and with skewness parameter beta = 1. A limit theorem for the maximal value of the sample {e(tXi), i = 1,...,N} is also proved.
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Copyright, Publisher and Additional Information:  (c) 2004, World Scientific. This is an author produced version of a paper published in Recent Developments in Stochastic Analysis and Related Topics. Uploaded in accordance with the publisher's selfarchiving policy. 
Keywords:  Random exponentials; Regular variation; Stable laws; Weak limit theorems; Sums of independent random variables; Kasahara's Tauberian theorem 
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Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) 
Depositing User:  Symplectic Publications 
Date Deposited:  16 Mar 2015 12:51 
Last Modified:  11 Aug 2015 13:29 
Published Version:  http://www.worldscientific.com/worldscibooks/10.11... 
Status:  Published 
Publisher:  World Scientific 
Refereed:  Yes 