Estimation of the mean of a stationary time series by sampling

Let X ( t ), – ∞ < t < ∞, be a stationary time series with mean c x . Let 0 < τ 1 < τ 2 < … < τ N ≦ T denote A given sampling times in the interval (0, T ]. We determine the asymptotic distribution of the estimate [ X (τ 1 ) + … + 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.