Ben Arous, G, Bogachev, LV and Molchanov, SA (2004) Limit laws for sums of random exponentials. In: Albeverio, S, Ma, Z-M and Roeckner, M, (eds.) Recent Developments in Stochastic Analysis and Related Topics. (1st Sino-German Conference on Stochastic Analysis, Beijing, China, Aug 29 - Sep 03, 2002). World Scientific , 45 - 65. ISBN 981-256-104-8
Abstract
We study the limiting distribution of the sum S-N(t) = Sigma(i=1)(N) e(tXi) as t -> infinity, N -> infinity, where (X-i) 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{X-I > 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.
Metadata
Item Type: | Book Section |
---|---|
Authors/Creators: |
|
Editors: |
|
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 self-archiving policy. |
Keywords: | Random exponentials; Regular variation; Stable laws; Weak limit theorems; Sums of independent random variables; Kasahara's Tauberian theorem |
Dates: |
|
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: | 21 Feb 2024 13:53 |
Published Version: | http://www.worldscientific.com/worldscibooks/10.11... |
Status: | Published |
Publisher: | World Scientific |
Refereed: | Yes |
Identification Number: | 10.1142/9789812702241_0004 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:83413 |