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Journal Article

On the Convergence of Moments in the Central Limit Theorem

Bengt Von Bahr
The Annals of Mathematical Statistics
Vol. 36, No. 3 (Jun., 1965), pp. 808-818
Stable URL: http://www.jstor.org/stable/2238192
Page Count: 11
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On the Convergence of Moments in the Central Limit Theorem
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Abstract

Let X1, X2, ⋯, Xn be a sequence of independent random variables (r.v.'s) with zero mean and finite standard deviation σi, 1 ≤ i ≤ n. According to the central limit theorem, the normed sum Yn = (1/sn) ∑n i=1 Xi, where sn = ∑n i=1 σ2 i, is under certain additional conditions approximatively normally distributed. We will here examine the convergence of the moments and the absolute moments of Yn 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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