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

• Published in: Statistics & probability letters Vol. 131; pp. 56 - 63
• Main Authors: Klesov, Oleg, Molchanov, Ilya
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2017
• Subjects: Multiindices; Multiple sums of random variables; Random measures; Strong law of large numbers; 60D05; 60G55; Multiple sums of random variables; Random measures; Strong law of large numbers; Multiindices; 60F15
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2017.08.007

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. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.