Asymptotics of randomly stopped sums in the presence of heavy tails

DOI10.3150/10-BEJ251zbMATH Open1208.60041arXiv0808.3697MaRDI QIDQ627282

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Publication date: 28 February 2011

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Abstract: We study conditions under which $P(S_{a}u>x)simP(M_{a}u>x)simEauP(x{i}_1>x)$ as $xoinfty$, where $S_{a}u$ is a sum $x{i}_1+\mathrm{...}+x{i}_{a}u$ of random size $au$ and $M_{a}u$ is a maximum of partial sums $M_{a}u=ma{x}_{nleau}{S}_n$. Here $x{i}_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $au$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $Exi>0$ and where the tail of $au$ is comparable with or heavier than that of $xi$, and obtain the asymptotics $P(S_{a}u>x)simEauP(x{i}_1>x)+P(au>x/Exi)$ as $xoinfty$. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $P(S_n>x)/P(x{i}_1>x)$ which substantially improve Kesten's bound in the subclass ${mathcalS}^{*}$ of subexponential distributions.

Full work available at URL: https://arxiv.org/abs/0808.3697

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