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Asymptotics of randomly stopped sums in the presence of heavy tails

Denisov, Denis, Foss, Serguei and Korshunov, Dmitry 2010. Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli -London- 16 (4) , pp. 971-994. 10.3150/10-BEJ251

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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 = maxn≤τ Sn. 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 of subexponential distributions.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: convolution equivalence; heavy-tailed distribution; random sums of random variables; subexponential distribution; upper bound
Additional Information: Pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/1350-7265/ (accessed 25/02/2014)
Publisher: Bernoulli Society for Mathematical Statistics and Probability
ISSN: 1350-7265
Date of First Compliant Deposit: 30 March 2016
Last Modified: 26 Jun 2019 01:56
URI: http://orca.cf.ac.uk/id/eprint/11067

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