Abstract
Forn≧1, letS n=ΣX n,i (1≦i≦r n <∞), where the summands ofS 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<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R.
Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.
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References
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W. Owen,An estimate for E|S n | for variables in the domain of normal attraction of a stable law of index α, 1<α<2. Ann. Probability1 (1973), 1071–1073.
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Berman, J. A limit theorem for the moments of sums of independent random variables. Israel J. Math. 31, 383–393 (1978). https://doi.org/10.1007/BF02761503
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DOI: https://doi.org/10.1007/BF02761503