MOMENT GENERATING FUNCTIONS OF QUADRATIC FORMS IN SERIALLY CORRELATED NORMAL VARIABLES

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Abstract
A method used by Kac in the study of Wiener functionals is adapted to the problem of calculating in closed form the joint moment generating functions of linear combinations of quadratic forms (not simultaneously diagonable) in serially correlated normal variables. A class of Gaussian processes is found for which this method is successful. The results are worked out in detail for the special case of the Uhlenbeck-Ornstein process, which includes first order stochastic difference equations with constant coefficient p. Exact moments of several estimates of the variance and autocorrelation are studied. Asymptotic results as the number n + 1 of observations → ∞ and the interval h between observations → 0 are derived under various assumptions on the limit of T = nh.

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