## Abstract

Let {X_{n}; n≥1} be a sequence of independent copies of a real-valued random variable X and set S_{n}=X_{1}+{dot operator}{dot operator}{dot operator}+X_{n}, n≥1. This paper is devoted to a refinement of the classical Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. We show that for 0 < p < 2, n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159-186, 1974) inequality to obtain some general results for sums of the form, (where {V_{n}; n≥1} is a sequence of independent Banach-space-valued random variables, and a_{n}≥0, n≥1), which may be of independent interest, but which we apply to.

Original language | English (US) |
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Pages (from-to) | 1130-1156 |

Number of pages | 27 |

Journal | Journal of Theoretical Probability |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2011 |

## Keywords

- Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers
- Rademacher type p Banach space
- Real separable Banach space
- Stable type p Banach space
- Sums of i.i.d. random variables