Asymptotic Behavior of the Variance of the Best Linear Unbiased Estimator for the Mean of a Discrete-time Singular Stationary Process

Abstract

It is known that for a wide class of discrete-time stationary processes possessing spectral densities f, the variance σ2(f) of the best linear unbiased estimator for the mean depends asymptotically only on the behavior of the spectral density f near the origin, and behaves hyperbolically as n → ∞. In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of σ2(f) as n → ∞. In particular, we show that a necessary condition for σ2(f) to decrease to zero exponentially is that the spectral density f vanishes on a set of positive measure in any vicinity of zero, and if f vanishes only at the origin, then it is impossible to obtain exponential decay of σ2(f), no mater how high the order of the zero of f at the origin.