Limit theorems for sums of random exponentials

Gérard Ben Arous, Leonid V. Bogachev, Stanislav A. Molchanov

Research output: Contribution to journalArticle

Abstract

We study limiting distributions of exponential sums SN(t) = ∑Ni=1e1xi as t→∞, N→∞, where (X i ) are i.i.d. random variables. Two cases are considered: (A) ess sup X i = 0 and (B) ess sup X i = ∞. 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 ∞ with index 1 < script Q sign <∞ (case B) or 0 < script Q sign < ∞ (case A). The appropriate growth scale of N relative to t is of the form eλH0(t) (0 < λ <) where the rate function H 0(t) is a certain asymptotic version of the function H (t)= log E[etXi](case B) or H (t)= -log E[etXi] (case A). We have found two critical points, λ12, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ2, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (script Q sign λ) ∈(0,2) and skewness parameter β ≡ 1.

Original languageEnglish (US)
Pages (from-to)579-612
Number of pages34
JournalProbability Theory and Related Fields
Volume132
Issue number4
DOIs
StatePublished - Jul 2005

Keywords

  • Central limit theorem
  • Exponential Tauberian theorems
  • Infinitely divisible distributions
  • Random exponentials
  • Regular variation
  • Stable laws
  • Sums of independent random variables
  • Weak limit theorems

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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