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

## TL;DR

This paper extends the strong law of large numbers to multiple sums with general normalizations, establishing new integrability conditions and applying results to ergodic theorems for random measures and point processes.

## Contribution

It generalizes the strong law of large numbers for multiple sums beyond i.i.d. variables and introduces new normalization techniques under broader conditions.

## Key findings

- Established equivalence between strong law validity and integrability of |Z| (log^+|Z|)^{r-1}.
- Proved a multiple sum generalization of the Brunk–Prohorov strong law of large numbers.
- Derived normalization conditions for sums with finite moments of order 2q.

## Abstract

The validity of the strong law of large numbers for multiple sums $S_n$ of independent identically distributed random variables $Z_k$, $k\leq n$, with $r$-dimensional indices is equivalent to the integrability of $|Z|(\log^+|Z|)^{r-1}$, 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 $q\geq1$, we show that the strong law of large numbers holds with the normalisation $\|n_1\cdots n_r\|^{1/2}(\log n_1\cdots\log n_r)^{1/(2q)+\varepsilon}$ for any $\varepsilon>0$. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.