A fresh look at distribution theory for quadratic forms in jointly normally distributed random variables

Abstract

Several major theorems pertaining to necessary and sufficient conditions under which quadratic forms in normal random vectors are chi-square distributed (both central and non central) are revisited and illuminated. Some of these results date back to the early 1950’s and are scattered among journals that are not all easily accessible. Moreover, some of these presentations are very terse, reflecting the tendency to reduce the amount of technical typesetting. Most of these results are not available in their entirety in current graduate level statistics texts. Accordingly, the purpose here is to collect these in one place and to offer complete detailed and clear proofs with some new perspectives. The general case in which the variance matrix of the normal random vector can be positive semidefinite is emphasized, since those results will subsume the positive definite case with appropriate reductions of conditions.