The Joint Moment Generating Function of Quadratic Forms in Multivariate Autoregressive Series

Let (<italic>X</italic>₁) be a discrete multivariate Gaussian autoregressive process of order 1. The paper derives the exact finite-sample joint moment generating function (m.g.f.) of the three quadratic forms constituting the sufficient statistic of the process. The formula is then specialized to some cases of interest, including the m.g.f. of functional of multivariate Ornstein-Uhlenbeck processes that arise asymptotically from more general (<italic>X</italic>₁) processes as well.