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Normal characterization by zero correlations

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  09 April 2009

Eugene Seneta  and
Gabor J. Szekely
Eugene Seneta
Affiliation:
School of Mathematics and Statistics University of SydneyNSW 2006Australia e-mail: eseneta@maths.usyd.edu.au
Gabor J. Szekely
Affiliation:
Department of Mathematics and Statistics Bowling Green State UniversityBowling Green OH 43403USA e-mail: gabors@bgnet.bgsu.edu
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Abstract

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Suppose Xi, i = 1,…,n are indepedent and identically distributed with E/X1/r < ∞, r = 1,2,…. If Cov (( − μ)r, S2) = 0 for r = 1, 2,…, where μ = EX1, S2 = , and , then we show X1 ~ N (μ, σ2), where σ2 = Var(X1). This covariance zero condition charaterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of and S2 using semi invariants. More generally, if Cov = 0 for r = 1,…, k, then E((X1 − μ)/σ)r+2 = EZr+2 for r = 1,… k, where Z ~ N(0, 1). Conversely Corr may be arbitrarily close to unity in absolute value, but for unimodal X1, Corr2( < 15/16, and this bound is the best possible.

MSC classification

Secondary: 60E05: Distributions 62E10: Characterization and structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 3 , December 2006 , pp. 351 - 361
Copyright
Copyright © Australian Mathematical Society 2006

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