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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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