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
Gabor J. Szekely
  Department of Mathematics and Statistics
  Bowling Green State University
  Bowling Green
  OH 43403

Suppose $X_i$, $i = 1,\dots,n$ 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 $S^2$ 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 $X_1$, $\operatorname{Corr}^2 ({\bar X}, S^2)<15/16$, and this bound is the best possible.
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