## Abstract

We study conditions under which P{S_{t} > x} ~ P{M _{t} > x} ~ EtP{?_{1} > x} as x?8, where S_{t} is a sum ?_{1}+ ? ?_{t} of random size t and Mt is a maximum of partial sums Mt = max_{n=t} S_{n}. Here, ?_{n}, n = 1, 2,. . ., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where t is independent of the summands; also, in a particular situation, we deal with a stopping time. We also consider the case where E? > 0 and where the tail of t is comparable with, or heavier than, that of ?, and obtain the asymptotics P{S_{t} > x} ~ EtP{?1 > x} + P{t > x/E?} as x?8. This case is of primary interest in branching processes. In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{S_{n} > x}/P{?1 > x} which substantially improve Kesten's bound in the subclass S* of subexponential distributions. © 2010 ISI/BS.

Original language | English |
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Pages (from-to) | 971-994 |

Number of pages | 24 |

Journal | Bernoulli |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2010 |

## Keywords

- Convolution equivalence
- Heavy-tailed distribution
- Random sums of random variables
- Subexponential distribution
- Upper bound