An application of Rosenthal's moment inequality to the strong law of large numbers (Q1069549)

Let \((X_ n)\) be a sequence of independent r.v.'s with \(E(X_ n)=0\) and \((a_ n)\) be a strictly increasing sequence in ]0,\(\infty [\). If for one \(p\geq 1\) (i) \(\sum_{n}a_ n^{-2p}E(| X_ n|^{2p})<\infty,\) (ii) \(\sum_{n}a_ n^{-2}(a^ 2_ n-a^ 2_{n-1})^{1-p}(E(X^ 2_ n))^ p<\infty\) then we have \(\lim a_ n^{-1}\sum^{n}_{i=1}X_ i=0\) a.s.. In the case \(a_ n=n\) this result is stronger than the classical strong laws of \textit{H. D. Brunk} [Duke Math. J. 15, 181-195 (1948; Zbl 0030.20003)] and \textit{K. L. Chung} [Proc. Berkeley Sympos. math. Statist. Probability 1950, 341-352 (1951; Zbl 0044.137)]. By a completely different and more complicated method we have recently replaced (i) by the weaker condition (i') \(\sum_{n}P\{| X_ n| >\epsilon a_ n\}<\infty\) \((\epsilon >o)\).