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Sibirskii Matematicheskii Zhurnal, 2004, Volume 45, Number 6, Pages 1401–1420
(Mi smj1150)
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This article is cited in 10 scientific papers (total in 10 papers)
Estimates for the distributions of the sums of subexponential random variables
V. V. Shneer Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
Let $\{S_n\}_{n\geqslant1}$ be a random walk with independent identically distributed increments $\{\xi_i\}_{i\geqslant1}$. We study the ratios of the probabilities $\mathbf{P}(S_n>x)/\mathbf{P}(\xi_1>1)$ for all $n$ and $x$. For some subclasses of subexponential distributions we find upper estimates uniform in $x$ for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence $\mathbf{P}(S_{\tau}>x)\sim\mathbf{E}_{\tau}\mathbf{P}(\xi_1>x)$ as $x\to\infty$. Here $\tau$ is a positive integer-valued random variable independent of $\{\xi_i\}_{i\geqslant1}$. The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.
Keywords:
subexponential distribution, distribution with long tail, distribution of dominated variation, sums of random variables, random walk, modulated random walk, supremum of random walk.
Received: 01.10.2003 Revised: 24.03.2004
Citation:
V. V. Shneer, “Estimates for the distributions of the sums of subexponential random variables”, Sibirsk. Mat. Zh., 45:6 (2004), 1401–1420; Siberian Math. J., 45:6 (2004), 1143–1158
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https://www.mathnet.ru/eng/smj1150 https://www.mathnet.ru/eng/smj/v45/i6/p1401
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Abstract page: | 351 | Full-text PDF : | 122 | References: | 41 |
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