On the estimation of the mean of a random process from irregular observations (Q2737037)

The author is interested in empirical estimation of the expectation of some stationary process \(x(t),\) \(t\in Z,\) with expectation \(m_{x},\) covariance function \(R_{x}(t)=\text{cov}(x(0),x(t)),\) \(t\in Z,\) and spectral density \(f_{x}(\lambda),\) \(\lambda\in \Pi=[-\pi,\pi].\) Two cases are considered: (i) the process \(x(t)\) has weak (or short) dependence, namely, \(\sum_{t\in Z}|R_{x}(t)|<\infty\); (ii) the process has strong (or long) dependence, namely, \(\sum_{t\in Z}|R_{x}(t)|=\infty.\) The aim of this paper is to estimate the unknown expectation \(m_{x}=Ex(t)\) of the random process \(x(t)\) from observations \(y(t)=x(t)d(t)\), \(t\in \{0,1,2,\dots,T-1\},\) where \(x(t)\) is a process with short or long memory and \(d(t)\) is the Bernoulli sequence. It is supposed that the observations are irregular or with gaps. The comparison for asymptotic variances of the sample expectation of the process with long dependence under irregular and regular observations is given.