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 |
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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: | 16 May 2023 00:50 |
URI: | https://orca.cardiff.ac.uk/id/eprint/11067 |
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