Order of growth of a random field (Q1821426)

The author considers some extensions of estimates in the strong law of large numbers obtained by \textit{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 \(n\to (\infty,...,\infty)\), \(\delta >0\).