The existence of moments of the suprema of multiple sums and the strong law of large numbers (Q2737021)

Let \(d\geq 1,\) and let \(N^{d}\) be a set of \(d\)-dimensional vectors with natural coordinates. Let us consider the field of i.i.d. random variables \(\{X(\overline n)\); \(\overline n\in N^d\}\) and the field of their sums \(S(\overline n)=\sum_{\overline k\leq\overline n}X(\overline k),\) where ``\(\leq\)'' means the coordinate order in \(N^{d}.\) The strong law of large numbers in this case is the following relation: \(\lim_{\max}\frac{S(\overline n)}{b(\overline n)}=0\) a.s., where \(\lim_{\max}\) is a limit as \(\max(n_{1},\dots ,n_{d})\to\infty,\) \(\overline n=(n_{1},\dots ,n_{d}),\) and \(\{b(\overline n)\), \(\overline n\in N^{d}\}\) is a field of non-random numbers. Let us also consider the following relation: NEWLINE\[NEWLINE\lim_{n_{1}\to\infty}\sup_{n_{2}\geq 1,\dots ,n_{d}\geq 1}\frac{|S(\overline n)|}{b(\overline n)}=0 \;\text{ a.s.}NEWLINE\]NEWLINE It is obvious that the first relation follows from the second one. But the inverse assertion is not obvious. The aim of the present paper is to prove the equivalence of these two relations for a special class of fields \(\{b(\overline n),\overline n\in N^{d}\}.\) Another problem discussed in the paper is to find the moment conditions for the existence of the expectation \(E[\sup_{\overline n\in N^{d}}(|S(\overline n)|/b(\overline n))]^q\).