### 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 |

### Keywords

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