Klesov, Oleg; Molchanov, Ilya
(2017).
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Moment conditions in strong laws of large numbers for multiple sums and random measures.
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Statistics and Probability Letters, 131, pp. 56-63.
Elsevier
10.1016/j.spl.2017.08.007

Text
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The validity of the strong law of large numbers for multiple sums Sn of independent identically

distributed random variables Zk , k ≤ n, with r-dimensional indices is equivalent to the integrability of |Z|(log+|Z|)r⁻¹, where Z is the generic summand. We consider the strong law of large numbers for more general normalizations, without assuming that the summands Zk are identically distributed, and prove a multiple sum generalization of the Brunk–Prohorov strong law of large numbers. In the case of identical finite moments of order 2q with integer q ≥ 1, we show that the strong law of large numbers holds with the normalization (n₁ · · · nr)1/2(log n₁ · · · log nr)1/(2q)+ε for any ε > 0. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science |

## UniBE Contributor: |
Molchanov, Ilya |

## Subjects: |
500 Science > 510 Mathematics |

## ISSN: |
1669-1676 |

## Publisher: |
Elsevier |

## Funders: |
[4] Swiss National Science Foundation |

## Language: |
English |

## Submitter: |
Ilya Molchanov |

## Date Deposited: |
20 Mar 2018 10:46 |

## Last Modified: |
05 Dec 2022 15:09 |

## Publisher DOI: |
10.1016/j.spl.2017.08.007 |

## BORIS DOI: |
10.7892/boris.109665 |

## URI: |
https://boris.unibe.ch/id/eprint/109665 |