Moment conditions in strong laws of large numbers for multiple sums and random measures

Oleg Klesov and Ilya Molchanov

Statistics & Probability Letters, 2017, vol. 131, issue C, 56-63

Abstract: 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(logn1⋯lognr)1∕(2q)+ε for any ε>0.

Keywords: Multiple sums of random variables; Multiindices; Strong law of large numbers; Random measures (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1016/j.spl.2017.08.007

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