Generalizing a theorem of Katz

The Katz theorem states that if X1,X2,… are i.i.d. random variables, Sn=X1+⋯+Xn, n≥1, and t≥1, then ∑n≥1nt−2P(|Sn|≥εn)<∞, ε>0, if and only if E|X1|t<∞ and EX1=0. Assuming only that X1,X2,… are pairwise independent, but not necessarily identically distributed, we give sufficient conditions for the convergence of the series ∑n≥1nt−2P(|Sn|≥εn), ε>0, when 1≤t<3. Then, if X1,X2,… are independent, the sequence {med(Xn):n≥1} is bounded and t≥1, we show that one of these conditions is also necessary.