Distribution of the Average Power of a Normal Time Series

The distribution of the sum of the squares of N correlated and normally distributed elements of a time series can be computed by numerical quadrature of a Laplace inversion integral involving the moment generating function (m.g.f.) of the sum. A method is presented for computing that m.g.f. for a stationary autoregressive moving-average (ARMA) process whose spectral density is a known rational function with $2n$ poles. It requires evaluating determinants of 2n × 2n and (2n + 1) × (2n + 1) matrices, which may be much smaller than the N × N covariance matrix of the time series. A second method is described that is based on the Kalman equations and applies to time series, possibly nonstationary, generated by a discrete-time linear system driven by normal random noise. A third method, utilizing the Levinson algorithm, applies when the time series is merely stationary.