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Moment conditions in strong laws of large numbers for multiple sums and random measures. (English) Zbl 1386.60176
Summary: The validity of the strong law of large numbers for multiple sums $$S_{\mathbf{n}}$$ of independent identically distributed random variables $$Z_{\mathbf{k}}$$, $$\mathbf{k}\leq \mathbf{n}$$, 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 $$Z_{\mathbf{k}}$$ 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 $$q \geq 1$$, we show that the strong law of large numbers holds with the normalization $$(n_1 \cdots n_r)^{1/2}(\log n_1 \cdots \log n_r)^{1/(2q)+\varepsilon}$$ 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.
##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F15 Strong limit theorems 60D05 Geometric probability and stochastic geometry
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