### Abstract

Let a_{j} be positive weight constants and X_{j} be independent non-negative random variables (j=1,2,…) and S_{n}(a)=∑_{i=1} ^{n}a_{i}X_{i}. If the X_{j} have the same relatively stable distribution, then under mild conditions there exist constants b_{n}→∞ such that W¯_{n}(a)=b_{n} ^{−1}S_{n}(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the X_{j} are not identically distributed, but rather the tail probability P(a_{j}X_{j}>x) is asymptotically proportional to a_{j}(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim inf_{n→∞}W¯_{n}(a)=1 and lim sup_{n→∞}W¯_{n}(a)=∞.

Original language | English |
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Pages (from-to) | 8-16 |

Number of pages | 9 |

Journal | Statistics and Probability Letters |

Volume | 142 |

DOIs | |

Publication status | Published - 1 Nov 2018 |