### Abstract

Let {S_{n}}_{n=1} be a random walk with independent identically distributed increments {?_{i}}_{i=1}. We study the ratios of the probabilities P(S_{n} > x)/P(?_{1} > 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_{t} > x) ~ Et P(?_{1} > x) as x ? 8. Here t is a positive integer-valued random variable independent of {?_{i}} _{i = 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.

Original language | English |
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Pages (from-to) | 1143-1158 |

Number of pages | 16 |

Journal | Siberian Mathematical Journal |

Volume | 45 |

Issue number | 6 |

DOIs | |

Publication status | Published - Nov 2004 |

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### Keywords

- distribution of dominated variation
- distribution with long tail
- modulated random walk
- random walk
- subexponential distribution
- sums of random variables
- supremum of random walk

### Cite this

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*Siberian Mathematical Journal*, vol. 45, no. 6, pp. 1143-1158. https://doi.org/10.1023/B:SIMJ.0000048931.70386.68

**Estimates for the distributions of the sums of subexponential random variables.** / Shneer, V. V.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Estimates for the distributions of the sums of subexponential random variables

AU - Shneer, V. V.

PY - 2004/11

Y1 - 2004/11

N2 - Let {Sn}n=1 be a random walk with independent identically distributed increments {?i}i=1. We study the ratios of the probabilities P(Sn > x)/P(?1 > 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(St > x) ~ Et P(?1 > x) as x ? 8. Here t is a positive integer-valued random variable independent of {?i} i = 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.

AB - Let {Sn}n=1 be a random walk with independent identically distributed increments {?i}i=1. We study the ratios of the probabilities P(Sn > x)/P(?1 > 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(St > x) ~ Et P(?1 > x) as x ? 8. Here t is a positive integer-valued random variable independent of {?i} i = 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.

KW - distribution of dominated variation

KW - distribution with long tail

KW - modulated random walk

KW - random walk

KW - subexponential distribution

KW - sums of random variables

KW - supremum of random walk

UR - http://www.scopus.com/inward/record.url?scp=10044230420&partnerID=8YFLogxK

U2 - 10.1023/B:SIMJ.0000048931.70386.68

DO - 10.1023/B:SIMJ.0000048931.70386.68

M3 - Article

VL - 45

SP - 1143

EP - 1158

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 6

ER -