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

Abstract

The validity of the strong law of large numbers for multiple sums S n of independent identically distributed random variables Z k , 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 Z k 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 (n 1 · · · n r ) 1/2 (log n 1 · · · log n r ) 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.