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Asymptotics of randomly stopped sums in the presence of heavy tails

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<mark>Journal publication date</mark>2010
<mark>Journal</mark>Bernoulli
Issue number4
Volume16
Number of pages24
Pages (from-to)971-994
Publication StatusPublished
<mark>Original language</mark>English

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 ${\mathcal{S}}^{*}$ of subexponential distributions.