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 semiinvariants. 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.
