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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
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2238192
Page Count: 11
You can always find the topics here!Topics: Integers, Polynomials, Mathematical moments, Standard deviation, Central limit theorem, Random variables, Distribution functions, Sine function, Integrands, Mathematical constants
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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).
The Annals of Mathematical Statistics © 1965 Institute of Mathematical Statistics