Geodesic
Asymptotics in the Baum–Katz formula for random fields
Matematičeskie zametki, Tome 79 (2006) no. 5, pp. 674-680 Cet article a éte moissonné depuis la source Math-Net.Ru

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In 1996, D. Deng established an analog of the Baum–Katz theorem on the convergence rate in the law of large numbers for multi-indexed random variables. The series describing the convergence rate depends, in a natural way, on the parameter characterizing the excess of the normalized sums over some level. In this paper, we find the precise asymptotics of the sum of this series with respect to the above-mentioned parameter. Thus, a generalization of a recent result due to A. Gut and A. Spataru is obtained. 
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S. V. Dil'man. Asymptotics in the Baum–Katz formula for random fields. Matematičeskie zametki, Tome 79 (2006) no. 5, pp. 674-680. http://geodesic.mathdoc.fr/item/MZM_2006_79_5_a3/

[1] Baum L. E., Katz M., “Convergence rate in the law of large numbers”, Trans. Amer. Math. Soc., 120 (1965), 108–123 | DOI | MR | Zbl

[2] Gut A., “Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices”, Ann. Probab., 6 (1978), 469–482 | DOI | MR | Zbl

[3] Deng D., “Complete convergence and convergence rates in Marcinkiewicz law of large numbers for random variables indexed by $\mathbb Z_+^d$”, Math. Appl., 9 (1996), 441–448 | MR | Zbl

[4] Chen R., “A remark on the tail probability of a distribution”, J. Multivariate Anal., 8 (1978), 328–333 | DOI | Zbl

[5] Gut A., Spataru A., “Precise asymptotics in the Baum–Katz and Davis law of large numbers”, J. Math. Anal. Appl., 248 (2000), 233–246 | DOI | Zbl

[6] Gut A., Spataru A., “Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables”, J. Multivariate Analysis, 86 (2003), 398–422 | DOI | Zbl