Blocks of orthogonal random variables and the strong law of large numbers (Q2367062)

Let \(\{X_ k\}\) be a sequence of random variables such that \(E(X_ k^ 2)=\sigma_ k^ 2<\infty\) and \(E(X_ k)=0\) (\(k=1,2,\dots\)). It is well-known that if (1) \(E(X_ k X_ l)=0\) \((k\neq l\); \(k,l=1,2,\dots)\) and (2) \(\sum_{k=1}^ \infty {{\sigma_ k^ 2} \over {k^ 2}}(\log k)^ 2<\infty\), then (3) \({1\over n}(X_ 1+\dots+X_ n)\to 0\) a.s. \((n\to\infty)\) [see \textit{J. L. Doob}, Stochastic processes (1953; Zbl 0053.268), p. 158]. This result remains true if (1) is weakened as follows (4) \(E(X_ k X_ l)=0\) \((2^{p-1}<k<l\leq 2^ p\); \(p,k,l=1,2,\dots)\) [\textit{F. Móricz}, Proc. Am. Math. Soc. 101, 709-715 (1987; Zbl 0632.60025)]. If we substitute in (4) the block \(p^ \alpha<k\leq (p+1)^ \alpha\) for \(2^{p-1}<k\leq 2^ p\), where \(\alpha>1\) is fixed, then (3) is no longer true; it is nevertheless true if we refine (2) as follows: (5) \(\sum_{k=1}^ \infty {{\sigma_ k^ 2} \over {k^{2- 1/\alpha}}}(\log k)^ 2<\infty\). This is our main result.