## Abstract

Let 0 < p < 2 and 1 ≤ q < ∞. Let {X_{n}; n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set S_{n} = X_{1} + · · ·+X_{n}, n ≥ 1. We say X satisfies the (p, q)-type strong law of large numbers (and write X ∈ SLLN(p, q)) if (formula given) < ∞ almost surely. This paper is devoted to a characterization of X ∈ SLLN(p, q). By applying results obtained from the new versions of the classical Lévy, Ottaviani, and Hoffmann- Jørgensen (1974) inequalities proved by Li and Rosalsky (2013) and by using techniques developed by Hechner (2009) and Hechner and Heinkel (2010), we obtain sets of necessary and sufficient conditions for X ∈ SLLN(p, q) for the six cases: 1 ≤ q < p < 2, 1 < p = q < 2, 1 < p < 2 and q > p, q = p = 1, p = 1 < q, and 0 < p < 1 ≤ q. The necessary and sufficient conditions for X ∈ SLLN(p, 1) have been discovered by Li, Qi, and Rosalsky (2011). Versions of the above results in a Banach space setting are also given. Illustrative examples are presented.

Original language | English (US) |
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Pages (from-to) | 539-561 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2016 |

### Bibliographical note

Publisher Copyright:© 2015 American Mathematical Society.

## Keywords

- (p-q)-type strong law of large numbers
- 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