On the lower tail probabilities of some random series

Research output

37 Citations (Scopus)

Abstract

The behavior of tail probabilities P{S ≤ r}, r → 0 is investigated, where S is a series S = ∑λjZj generated by some sequence of positive numbers {λj} and by a sequence {Zj} of independent copies of a positive random variable Z. We present the exact asymptotic expression for P{S ≤ r} by means of Laplace transform Λ(γ) = Eexp{-γS} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.

Original languageEnglish
Pages (from-to)424-442
Number of pages19
JournalAnnals of Probability
Volume25
Issue number1
DOIs
Publication statusPublished - 1 Jan 1997

Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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