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 , are independent and identically distributed with , . If for , where , , and , then we show that , where . 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 and using semi-invariants. More generally, if Cov for , then for , where . Conversely may be arbitrarily close to unity in absolute value, but for unimodal , , and this bound is the best possible.