Strong consistency with rates of spectral estimation of continuous-time processes: from periodic and poisson sampling schemes

Abstract

Let X = {X(t)} t∈ℝ be a continuous-time strictly stationary and strongly mixing process. In this article, we first prove the uniform complete convergence of the spectral density estimate from periodic sampling. Because of aliasing, however, this result requires strong conditions on the spectral density ϕ X . To overcome aliasing, we consider the sampled process {X tn } n∈ℤ , where {t n } is a stationary point process independent of X. The uniform complete convergence of the spectral density estimate based on the discrete time observation {X tk , t k } is also obtained.