zbMATH — the first resource for mathematics

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.
60G60 Random fields
Full Text: DOI
[1] L. E. Baum and M. Katz, ”Convergence rate in the law of large numbers,” Trans. Amer. Math. Soc., 120 (1965), 108–123. · Zbl 0142.14802 · doi:10.1090/S0002-9947-1965-0198524-1
[2] A. Gut, ”Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices,” Ann. Probab., 6 (1978), 469–482. · Zbl 0383.60030 · doi:10.1214/aop/1176995531
[3] D. Deng, ”Complete convergence and convergence rates in the Marcinkiewicz law of large numbers for random variables indexed by \(\mathbb{Z}\) + d ,” Math. Appl., 9 (1996), 441–448. · Zbl 0946.60027
[4] R. Chen, ”A remark on the tail probability of a distribution,” J. Multivariate Anal., 8 (1978), 328–333. · Zbl 0376.60033 · doi:10.1016/0047-259X(78)90084-2
[5] A. Gut and A. Spataru, ”Precise asymptotics in the Baum–Katz and Davis law of large numbers,” J. Math. Anal. Appl., 248 (2000), 233–246. · Zbl 0961.60039 · doi:10.1006/jmaa.2000.6892
[6] A. Gut and A. Spataru, ”Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables,” J. Multivariate Analysis, 86 (2003), 398–422. · Zbl 1031.60032 · doi:10.1016/S0047-259X(03)00050-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.