A limit theorem for the moments of sums of independent random variables

Forn≧1, letS n=ΣX n,i (1≦i≦r n S n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some $$\alpha > 0, f(x) \leqq f(tx) \leqq t^a f(x)$$ for allt≧1 and all values ofx. Theorem.For centering constants c n,let S n − c n converge in distribution to a random variable S. (A)In order that Ef(Sn − cn) converge to a limit L, it is necessary and sufficient that there exist a common limit $$R = \mathop {lim}\limits_{l \to \infty } \mathop {\underline {\overline {lim} } }\limits_{n \to \infty }^{} \sum\limits_{i - 1}^{r_n } {} \int {f(X_{n,i}^{} )I(|X_{n,i}^{} | > t).} $$ (B)If L exists, then L. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.