Asymptotics of randomly stopped sums in the presence of heavy tails

Denis Denisov, Serguei Foss, Dmitry Korshunov

Research output: Contribution to journalArticle

25 Citations (Scopus)

Abstract

We study conditions under which P{St > x} ~ P{M t > x} ~ EtP{?1 > x} as x?8, where St is a sum ?1+ ? ?t of random size t and Mt is a maximum of partial sums Mt = maxn=t Sn. Here, ?n, n = 1, 2,. . ., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where t is independent of the summands; also, in a particular situation, we deal with a stopping time. We also consider the case where E? > 0 and where the tail of t is comparable with, or heavier than, that of ?, and obtain the asymptotics P{St > x} ~ EtP{?1 > x} + P{t > x/E?} as x?8. This case is of primary interest in branching processes. In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{Sn > x}/P{?1 > x} which substantially improve Kesten's bound in the subclass S* of subexponential distributions. © 2010 ISI/BS.

Original languageEnglish
Pages (from-to)971-994
Number of pages24
JournalBernoulli
Volume16
Issue number4
DOIs
Publication statusPublished - Nov 2010

Keywords

  • Convolution equivalence
  • Heavy-tailed distribution
  • Random sums of random variables
  • Subexponential distribution
  • Upper bound

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