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Order of growth of a random field. (English. Russian original) Zbl 0616.60047

Math. Notes 39, 237-240 (1986); translation from Mat. Zametki 39, No. 3, 431-437 (1986).
The author considers some extensions of estimates in the strong law of large numbers obtained by V. V. Petrov [Theor. Probab. Appl. 14, 183–192 (1969); translation from Teor. Veroyatn. Primen. 14, 193–202 (1969; Zbl 0191.47602); Theory Probab. Appl. 18, 348–350 (1973); translation from Teor. Veroyatn. Primen. 18, 358–361 (1973; Zbl 0295.60020)] on the case of \(r\)-parameter field of random variables. For example, if \(\{X_ n\}_{n\in {\mathbb{N}}^ r}\) is a field of independent random variables with \(EX^ 2_ n\to \infty\), \(B_ n=\sum_{i<n}DX_ i\to \infty\) when \(n\to (\infty,...,\infty)\), \(E_ n=\{k\in N^ r:\) \(B_ k\leq B_ n\}\), then \[ S_ n-ES_ n=o((\log \sum_{k\in E_ n}DX_ k)^{1/2}+\delta)\quad a.s. \] when \(n\to (\infty,...,\infty)\), \(\delta >0\).
Reviewer: A. I. Ponomarenko

MSC:

60G60 Random fields
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[1] V. V. Petrov, ?On the strengthened law of large numbers,? Teor. Veroyatn. Primen.,14, No. 2, 193-202 (1969). · Zbl 0191.47602
[2] V. V. Petrov, ?On the order of growth of sums of dependent random variables,? Teor. Veroyatn. Primen.,18, No. 2, 358-361 (1973).
[3] V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag (1975). · Zbl 0322.60043
[4] A. I. Martikainen, ?On the iterated logarithm law for a random field,? Vestn. Leningr. Gos. Univ., Mat., Mekh., Astron.,22, 14-21 (1985).
[5] O. I. Klesov, ?The iterated logarithm law for multiple sums,? Teor. Veroyatn. Mat. Statistika, Kiev,27, 60-67 (1982). · Zbl 0504.60039
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