On the strong law of large numbers for sums of pairwise independent random variables (Q1907885)

Consider a sequence \(X_1, X_2, \dots\) of pairwise independent and identically distributed random variables. Let \(S_0= 0\), \(S_n= X_1+ \dots+ X_n\) \((n\geq 1)\) denote their corresponding partial sums. The author proves the following (improved) version of a strong law of large numbers: If \(E|X_1 |^\gamma (\log^+ |X_1 |)^\tau< \infty\), where \(1< \gamma< 2\), \(\tau> 0\), and \(\tau> 4\gamma- 6\), then \((S_n- ES_n )/ n^{1/ \gamma}\to 0\) a.s. as \(n\to \infty\).