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Asymptotics in the Baum-Katz formula for random fields. (English. Russian original) Zbl 1141.60026
Math. Notes 79, No. 5, 625-631 (2006); translation from Mat. Zametki 79, No. 5, 674-680 (2006).
30 years ago the reviewer extended the classical Baum-Katz theorem on convergence rates in the law of large numbers to random fields, more particulary, to random variables indexed by $$\mathbb{Z}^+_d$$, the positive integer lattice points. The probabilities involved were of the form $$P(|S_{{\mathbf n}}|> \varepsilon|{\mathbf n}|^\alpha)$$, where $${\mathbf n}= (n_1,n_2,\dots, n_d)$$ and $$|{\mathbf n}|^\alpha$$ is to interpreted as $$\prod^d_{k=1} n^\alpha_k$$. This result was extended by the author to the case when the coordinates were raised to different powers, viz., $$|{\mathbf n}|^\alpha$$ is replaced by $$\prod^d_{k=1} n^{\alpha_k}_k$$ under certain conditions.
Another aspect on convergence rates is to investigate the rate of convergence to infinity of the sum in terms of $$\varepsilon$$ as $$\varepsilon\to 0$$. For the classical Baum-Katz case this was done by the reviewer and Spătaru, and for the generalized case this is done in the paper under review.
##### MSC:
 60G60 Random fields
Full Text:
##### References:
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