The error variance of the optimal linear smoother and maximum-variance fractional pole models

The variance of the optimal one-step ahead linear prediction error of a discrete-time stationary stochastic process is given by the well-known Szegö-Kolmogorov formula as the geometric mean of the spectral density function. We first derive an analogous expression for the optimal linear smoother which uses the infinite past and the infinite future to determine the present. The least variance turns out to be the harmonic mean of the spectral density function. Building on this, we explore the question of what is the most random power spectrum in the sense of corresponding to the largest variance optimal linear smoother (i.e., least "smoothable"), which is consistent with finitely many covariance moments. It turns out that it can be described by an all-pole model, albeit the poles are fractional.