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 $S_n$ of independent identically distributed random variables $Z_k$, $k\le n$, with $r$-dimensional indices is equivalent to the integrability of $\mathrm{\mid }Z\mathrm{\mid }({\log }^{+}\mathrm{\mid }Z\mathrm{\mid }{)}^{r-1}$, where $Z$ is the typical summand. We consider the strong law of large numbers for more general normalisations, without assuming that the summands $Z_k$ are identically distributed, and prove a multiple sum generalisation of the Brunk–Prohorov strong law of large numbers. In the case of identical finite moments of irder $2q$ with integer $q\ge 1$, we show that the strong law of large numbers holds with the normalisation $|n_1\cdots n_r|^{1/2}(\log n_1\cdots\log n_r)^{1/(2q)+\varepsilon}$forany$\varepsilon>0$. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.