Bernoulli

  • Bernoulli
  • Volume 16, Number 4 (2010), 971-994.

Asymptotics of randomly stopped sums in the presence of heavy tails

Denis Denisov, Serguei Foss, and Dmitry Korshunov

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Abstract

We study conditions under which

P{Sτ > x} ∼ P{Mτ > x} ∼ EτP{ξ1 > x}  as x → ∞,

where Sτ is a sum ξ1 + ⋯ + ξτ of random size τ and Mτ is a maximum of partial sums Mτ = maxnτSn. Here, ξn, n = 1, 2, …, are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where τ is independent of the summands; also, in a particular situation, we deal with a stopping time.

We also consider the case where Eξ > 0 and where the tail of τ is comparable with, or heavier than, that of ξ, and obtain the asymptotics

P{Sτ > x} ∼ EτP{ξ1 > x} + P{τ > x / Eξ}  as x → ∞.

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{ξ1 > x} which substantially improve Kesten’s bound in the subclass ${\mathcal{S}}^{*}$ of subexponential distributions.

Article information

Source
Bernoulli Volume 16, Number 4 (2010), 971-994.

Dates
First available in Project Euclid: 18 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1290092892

Digital Object Identifier
doi:10.3150/10-BEJ251

Mathematical Reviews number (MathSciNet)
MR2759165

Zentralblatt MATH identifier
1208.60041

Keywords
convolution equivalence heavy-tailed distribution random sums of random variables subexponential distribution upper bound

Citation

Denisov, Denis; Foss, Serguei; Korshunov, Dmitry. Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli 16 (2010), no. 4, 971--994. doi:10.3150/10-BEJ251. https://projecteuclid.org/euclid.bj/1290092892.


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