Summing subsequences of random variables (Q579741)

For \(s=(s_ 1,...,s_ k)\in {\mathbb{N}}^ k\) with \({\mathbb{N}}=\{1,2,...\}\), put \(| s| =s_ 1...s_ k\) and \(r\leq s\) if \(r_ i\leq s_ i\), \(i=1,...,k\). Let \(X_ n\equiv X(n)\), \(n\in {\mathbb{N}}\), be a sequence of random variables satisfying only \(E| X_ n| (\log^+| X_ n|)^{k-1}\leq K<\infty\), \(n\in {\mathbb{N}}.\) It is proved that there is a strictly increasing sequence \(N=(n(1),n(2),...)\) such that for every subsequence \(N_ 1=(n(i_ 1),n(i_ 2),...)\) and every bijection \(f:N_ 1\to {\mathbb{N}}^ k\) with \(| f(n(i_ k))| \leq i_ k\), \(k\in {\mathbb{N}}\), we have \(| s|^{-1}\sum_{r\leq s}X(f^{-1}(r))\to X\) a.s., with \(E| X| <\infty\) and X not depending on \(N_ 1\) or f. This theorem extends and strengthens (to allow permutations) a result by Komlós, see \textit{M. Schwartz}, Acta Math. Hung. 47, 181-185 (1986; Zbl 0612.60032), and has connections with \textit{S. D. Chatterji}'s subsequence principle, Jahresber. Dtsch. Math.-Ver. 87, 91-107 (1985; Zbl 0571.60028).