Estimation of the mean of a stationary time series by sampling

Abstract

LetX(t), – ∞ <t< ∞, be a stationary time series with meancx. Let 0 <τ1<τ2 < … <τN≦Tdenote A given sampling times in the interval (0,T]. We determine the asymptotic distribution of the estimate [X(τ1) + … +X(τN)]/Nofcxwhen the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the seriesX(t). The results obtained may be viewed as non-standard central limit theorems.