Asymptotic behavior of the variance of the BLUE for the mean of stationary processes

TL;DR

This paper reviews how the variance of the BLUE for the mean of stationary processes behaves asymptotically, depending on spectral properties and model determinism.

Contribution

It provides a comprehensive survey of the asymptotic variance behavior of the BLUE, highlighting the influence of spectral density near zero and deterministic versus nondeterministic models.

Findings

Variance of BLUE exhibits hyperbolic behavior in nondeterministic models.

Variance decreases exponentially in purely deterministic models.

Spectral density vanishes on a set of positive measure near zero for exponential decay.

Abstract

In this paper, we survey results on the asymptotic behavior of the variance of the best linear unbiased estimator (BLUE) for the mean of stationary processes. This behavior is influenced by the regularity and memory structures of the observed models. The results show that the asymptotic behavior of the variance of the BLUE is determined solely by the behavior of the spectrum near the origin. For nondeterministic models, the variance of the BLUE exhibits hyperbolic behavior, similar to the power function, while for purely deterministic models, the variance decreases at an exponential rate. Specifically, a necessary condition for the variance of the BLUE to approach zero exponentially is that the spectral density of the model vanishes on a set of positive Lebesgue measure in any neighborhood of zero. We also present results on the asymptotic efficiency of various unbiased linear estimators in comparison to the BLUE.