## Abstract

Let {X, X_{i}, i = 1, 2, . . . } be independent nonnegative random variables with common distribution function F(x), and let N be an integer-valued random variable independent of X. Using S_{0} = 0 and S_{n} = S_{n} _{−1} + X_{n}, the random sum S_{N} has the distribution function (formula presented) and tail distribution Ḡ(x) = 1−G(x). Under suitable conditions, it can be proved that Ḡ(x) ∼ E(N)F(x) as x → ∞. In this paper, we extend previous results to obtain general bounds and asymptotic bounds and equalities for random sums where the components can be independent with infinite mean, regularly varying with index 1 or O-regularly varying. In the multivariate case, we obtain asymptotic equalities for multivariate sums with unequal numbers of terms in each dimension.

Original language | English |
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Article number | A009 |

Pages (from-to) | 433-450 |

Number of pages | 18 |

Journal | Lithuanian Mathematical Journal |

Volume | 55 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2015 |