Ben Arous, G, Bogachev, LV and Molchanov, SA (2005) Limit theorems for sums of random exponentials. Probability Theory and Related Fields, 132 (4). 579  612. ISSN 01788051
Abstract
We study limiting distributions of exponential sums $S_N(t)=\sum_{i=1}^N e^{tX_i}$ as $t\to\infty$, $N\to\infty$, where $(X_i)$ are i.i.d.\ random variables. Two cases are considered: (A) $\esssup X_i=0$ and (B) $\esssup X_i=\infty$. We assume that the function $h(x)=\log P(X_i>x)$ (case B) or $h(x)=\log P(X_i>1/x)$ (case A) is regularly varying at $\infty$ with index $1<\varrho<\infty$ (case B) or $0<\varrho<\infty$ (case A). The appropriate growth scale of $N$ relative to $t$ is of the form $e^{\lambda H_0(t)}$ ($0<\lambda<\infty$), where the rate function $H_0(t)$ is a certain asymptotic version of the function $H(t)=\log E [e^{tX_i}]$ (case B) or $H(t)=\log E [e^{tX_i}]$ (case A). We have found two critical points, $\lambda_{1}<\lambda_{2}$, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For $0<\lambda<\lambda_{2}$, under the slightly stronger condition of normalized regular variation of $h$ we prove that the limit laws are stable, with characteristic exponent $\alpha=\alpha(\varrho,\lambda)\in(0,2)$ and skewness parameter $\beta\equiv1$.
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Copyright, Publisher and Additional Information:  (c) 2005, Springer. This is an author produced version of a paper published in Probability Theory and Related Fields. Uploaded in accordance with the publisher's selfarchiving policy. The final publication is available at Springer via http://dx.doi.org/10.1007/s0044000404063 
Keywords:  central limit theorem; Stable laws; Weak limit theorems; Sums of independent random variables; Random exponentials; Regular variation; Exponential Tauberian theorems; Infinitely divisible distributions 
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Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) 
Depositing User:  Symplectic Publications 
Date Deposited:  22 May 2015 12:35 
Last Modified:  18 Jan 2018 12:47 
Published Version:  http://www.springerlink.com/content/gtayv8btrul1fl... 
Status:  Published 
Publisher:  SpringerVerlag 
Refereed:  Yes 
Identification Number:  https://doi.org/10.1007/s0044000404063 
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