Asymptotics in the Baum-Katz formula for random fields. (English. Russian original) Zbl 1141.60026

Dil’man, S. V.

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.

Reviewer: Allan Gut (Uppsala)

MSC:

60G60

Random fields

Keywords:

random field; independent identically distributed random variables; law of large numbers; normal distribution; precise asymptotics; Baum-Katz formula

References:

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