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 [summation operator]n>=1nt-2P(Sn>=[epsilon]n)<[infinity], [epsilon]>0, if and only if EX1t<[infinity] 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 [summation operator]n>=1nt-2P(Sn>=[epsilon]n), [epsilon]>0, when 1<=t<3. Then, if X1,X2,... are independent, the sequence is bounded and t>=1, we show that one of these conditions is also necessary.