Series representations of distributions of quadratic form in the normal vectors and generalised variance

Let the column vectors of X: p - n be distributed as independent normals with the same covariance matrix [Sigma]. Then, the quadratic form in normal vectors is denoted by XAX' = S, where A: n - n is a symmetric matrix which is assumed to be positive definite. This paper deals with the various series representations of the density function of S when E(X) = 0, extending the idea of Kotz et al. (1967) to the multivariate case. Further, it gives the distribution of S when E(X) [not equal to] 0, and the results for the univariate distribution of the quadratic form in noncentral normal variates can be obtained by putting p = 1.