Estimates for the distributions of the sums of subexponential random variables

Research output: Contribution to journalArticle

Abstract

Let {Sn}n=1 be a random walk with independent identically distributed increments {?i}i=1. We study the ratios of the probabilities P(Sn > x)/P(?1 > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(St > x) ~ Et P(?1 > x) as x ? 8. Here t is a positive integer-valued random variable independent of {?i} i = 1 The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.

Original languageEnglish
Pages (from-to)1143-1158
Number of pages16
JournalSiberian Mathematical Journal
Volume45
Issue number6
DOIs
Publication statusPublished - Nov 2004

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Subexponential Distribution
Random walk
Random variable
Regenerative Process
Asymptotic Equivalence
Uniform Estimates
Independent Random Variables
Identically distributed
Estimate
Increment
Tail
Integer
Sufficient Conditions

Keywords

  • distribution of dominated variation
  • distribution with long tail
  • modulated random walk
  • random walk
  • subexponential distribution
  • sums of random variables
  • supremum of random walk

Cite this

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abstract = "Let {Sn}n=1 be a random walk with independent identically distributed increments {?i}i=1. We study the ratios of the probabilities P(Sn > x)/P(?1 > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(St > x) ~ Et P(?1 > x) as x ? 8. Here t is a positive integer-valued random variable independent of {?i} i = 1 The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.",
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Estimates for the distributions of the sums of subexponential random variables. / Shneer, V. V.

In: Siberian Mathematical Journal, Vol. 45, No. 6, 11.2004, p. 1143-1158.

Research output: Contribution to journalArticle

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