ON THE EXISTENCE OF MOMENTS OF RATIOS OF QUADRATIC-FORMS

We obtain simple and generally applicable conditions for the existence of mixed moments E([X'AX](u)/[X'BX](upsilon)) of the ratio of quadrat ic forms T = X'AX/X'BX, where A and B are n x n symmetric matrices and X is a random n-vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. Th e result applies to the ratio of multivariate quadratic polynomials an d can be expected to remain valid in most situations in which X is sub ject to linear constraints. If u less than or equal to upsilon, the pr ecise distribution of X, and in particular the existence of moments of X, is virtually irrelevant to the existence of the mixed moments of T ; if u > upsilon, a prerequisite for existence of the (u, upsilon)th m ixed moment is the existence of the 2(u - upsilon)th moment of X. When X is not degenerate, the principal further requirement for the existe nce of the mixed moment is that B has rank exceeding 2 upsilon.