Abstract
Let \(\left\{ {S_n } \right\}_{n \geqslant 1} \) be a random walk with independent identically distributed increments \(\left\{ {\xi _i } \right\}_{i \geqslant 1} \). We study the ratios of the probabilities P(S n >x) / P(ξ1 > x) 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 P(S τ > x) ∼ E τ P(ξ1 > x) as x → ∞. Here τ is a positive integer-valued random variable independent of \(\left\{ {\xi _i } \right\}_{i \geqslant 1} \). 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.
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Shneer, V.V. Estimates for the Distributions of the Sums of Subexponential Random Variables. Siberian Mathematical Journal 45, 1143–1158 (2004). https://doi.org/10.1023/B:SIMJ.0000048931.70386.68
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DOI: https://doi.org/10.1023/B:SIMJ.0000048931.70386.68