Some characteristic and noncharacteristic properties of the Wishart distribution

Abstract

If the elements of a matrix $S$ follow a central Wishart distribution $W_{k}(n,\Sigma)$ and $a'\Sigma a\neq 0$ it is wellknown that $a'Sa/a'\Sigma a$ is distributed as a chi-square on $\nu$ d.f.. Further $a'\Sigma a=1 \Longrightarrow a'Sa=0$ with probability one. The object of this paper is to show through a counter-example that the converse of this result is not necessarily true unless $\Sigma$ is of rank one. It is shown, however, that if for every matrix $L$, such that $L\Sigma L'=I$, the diagonal elements of $LSL'$ are distributed as independent chisquare variables on $\nu$ d.f., then $S$ has a central Wishart distribution $W_{k}(\nu,\Sigma)$.