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June, 1965 On the Convergence of Moments in the Central Limit Theorem
Bengt Von Bahr
Ann. Math. Statist. 36(3): 808-818 (June, 1965). DOI: 10.1214/aoms/1177700055


Let $X_1, X_2, \cdots, X_n$ be a sequence of independent random variables (r.v.'s) with zero mean and finite standard deviation $\sigma_i, 1 \leqq i \leqq n$. According to the central limit theorem, the normed sum $Y_n = (1/s_n) \sum^n_{i=1} X_i,$ where $s_n = \sum^n_{i=1} \sigma^2_i$, is under certain additional conditions approximatively normally distributed. We will here examine the convergence of the moments and the absolute moments of $Y_n$ towards the corresponding moments of the normal distribution. The results in this general case are stated in Theorem 3 and Theorem 4, but, in order to avoid repetition and unnecessary complication, explicit proofs will only be given in the case of equally distributed random variables. (Theorem 1 and Theorem 2).


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Bengt Von Bahr. "On the Convergence of Moments in the Central Limit Theorem." Ann. Math. Statist. 36 (3) 808 - 818, June, 1965.


Published: June, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0139.35301
MathSciNet: MR179827
Digital Object Identifier: 10.1214/aoms/1177700055

Rights: Copyright © 1965 Institute of Mathematical Statistics

Vol.36 • No. 3 • June, 1965
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