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A Property of the Normal Distribution
Eugene Lukacs and Edgar P. King
The Annals of Mathematical Statistics
Vol. 25, No. 2 (Jun., 1954), pp. 389-394
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2236741
Page Count: 6
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The following theorem is proved. Let X1, X2, ⋯, Xn be n independently (but not necessarily identically) distributed random variables, and assume that the nth moment of each Xi(i = 1, 2, ⋯, n) exists. The necessary and sufficient conditions for the existence of two statistically independent linear forms Y1 = ∑n s=1 asX s and Y2 = ∑n s=1bsX s are: (A) Each random variable which has a nonzero coefficient in both forms is normally distributed. (B) ∑n s=1asb sσ2 s = 0. Here σ2 s denotes the variance of Xs (s = 1, 2, ⋯, n). For n = 2 and a1 = b1 = a2 = 1, b2 = -1 this reduces to a theorem of S. Bernstein . Bernstein's paper was not accessible to the authors, whose knowledge of his result was derived from a statement of S. Bernstein's theorem contained in a paper by M. Frechet . A more general result, not assuming the existence of moments was obtained earlier by M. Kac . A related theorem, assuming equidistribution of the Xi (i = 1, 2, ⋯ n) is stated without proof in a recent paper by Yu. V. Linnik .
The Annals of Mathematical Statistics © 1954 Institute of Mathematical Statistics