The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics

Abstract

The following theorem is proved. If a univariate distribution has moments of first and second order and admits a homogeneous and symmetric quadratic statistic $Q$ which is independently distributed of the mean of a sample of $n$ drawn from this distribution, then it is either the normal distribution ($Q$ is then proportional to the variance) or the degenerate distribution (in this case no restriction is imposed on $Q$) or a step function with two symmetrically located steps (in this case $Q$ is the sum of the squared observations). The converse of this statement is also true.