On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions

Let {an,n≥1} be a sequence of positive constants with an/n↑ and let {X,Xn,n≥1} be a sequence of pairwise independent identically distributed random variables. In this paper, we obtain the strong law of large numbers and complete convergence for the sequence {X,Xn,n≥1}, which are equivalent to the general moment condition ∑n=1∞P(|X|>an)<∞. We obtain, as a corollary, the strong law of large numbers due to Kruglov [Kruglov, V.M., 2008. A strong law of large numbers for pairwise independent identically distributed random variables with infinite means. Statist. Probab. Lett. 78, 890–895].