The law of the iterated logarithm for multiple sums (Q2740443)

Let \(\{X(\bar n),\bar n\in N^{d}\}\) be a field of i.i.d. r.v., \(EX=0, EX^{2}=1,\) and let \((S(\bar n), \bar n\in N^{d})\) be a field of partial sums, namely, \(S(\bar n)=\sum_{\bar k\leq\bar n}X(\bar k),\) where \(\leq\) is an order by coordinate in the space \(N^{d}.\) For such weighted multiple sums of i.i.d. r.v. the law of the iterated logarithm (LIL) is proved. The LIL for sequences of partial sums of i.i.d. r.v. was proved by \textit{P. Hartman} and \textit{A. Winter} [Am. J. Math. 63, 169-176 (1941; Zbl 0024.15802)] under the conditions \(EX=0\) and \(EX^{2}=1.\) \textit{V. Strassen} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 4, 265-268 (1966; Zbl 0141.16501)] proved that from the LIL the latter conditions follow. For the case of the field of partial sums of the i.i.d. r.v., \textit{M. J. Wichura} [Ann. Probab. 1, 272-296 (1973; Zbl 0288.60030)] proved that the LIL is equivalent to the latter conditions as \(d\geq 2\) and \(E(X^{2}((\ln^{+}|X|)^{d-1})/(\ln^{+}\ln^{+}|X|))<\infty\).