Normal characterization by zero correlations

J. Aust. Math. Soc. 81 (2006), 351-361

Normal characterization by zero correlations

Eugene Seneta
School of Mathematics and Statistics
University of Sydney
NSW 2006
Australia
eseneta@maths.usyd.edu.au

and

Gabor J. Szekely
Department of Mathematics and Statistics
Bowling Green State University
Bowling Green
OH 43403
USA
gabors@bgnet.bgsu.edu

Abstract

Suppose

, $i = 1,\dots,n$ are independent and identically distributed with $E{|X_1|}^r < \infty$ , $r = 1,2,\ldots$ . If $\operatorname{Cov} \big({({\bar X} - \mu)}^r, S^2\big) = 0$ for $r = 1, 2, \dots$ , where $\mu = E X_1$ , $S^2 = \sum_{i=1}^n {(X_i - {\bar X})^2}/{(n-1)}$ , and ${\bar X} = \sum_{i=1}^n {X_i}/{n}$ , then we show that $X_1 \sim {\mathcal{N}} (\mu, \sigma^{2})$ , where $\sigma^2 = \operatorname{Var} (X_1)$ . This covariance zero condition characterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of $\bar X$ and

using semi-invariants. More generally, if Cov $\operatorname{Cov} ({({\bar X} - \mu)}^r, S^2) = 0$ for $r = 1,\dots,k$ , then $E((X_1 - \mu)/\sigma)^{r+2} = EZ^{r+2}$ for $r = 1,\dots,k$ , where $Z \sim {\mathcal{N}} (0, 1)$ . Conversely $\operatorname{Corr} ({({\bar X}-\mu)}^r,S^2)$ may be arbitrarily close to unity in absolute value, but for unimodal

, $\operatorname{Corr}^2 ({\bar X}, S^2)<15/16$ , and this bound is the best possible.

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