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

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.