3.13.3 Moments of a Quadratic Polynomial of a Multinormal Random Vector

3.13.3 Moments

We have seen that a random variable Y that is a quadratic polynomial of a joint-normal random vector X ~ Nm(μ,Σ) can be expressed as a linear polynomial of independent chi-squared and normal random variables. Based upon this representation, we may apply [3.143] to obtain the MGF of Y. From this, we can calculate the moments of Y. The details of the derivation are covered by Mathai and Provost (1992). Results, based upon notation introduced earlier in this section, are as follows.

Define, for positive integers k,


where any undefined term 00 is set equal to 0. Employing notation introduced in Section 3.8.2, the rth moment of Y is


where any empty product is interpreted as equaling 1. Based upon [3.182],






and so forth according to a similar pattern.