Estimation of the mean of a stationary time series by sampling

Abstract

Let X(t), – ∞ < t < ∞, be a stationary time series with mean c_x. Let 0 < τ₁ < τ2 < … < τ_N ≦ T denote A given sampling times in the interval (0, T]. We determine the asymptotic distribution of the estimate [X(τ₁) + … + X(τ_N)]/N of c_x when 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 series X(t). The results obtained may be viewed as non-standard central limit theorems.