Discrete Periodic Sampling with Jitter and Almost Periodically Correlated Processes

Abstract

The zero-mean process $\{X(t){:}\ t\in\mathbb{R} \}$ is said to be almost periodically correlated whenever its shifted covariance kernel ${(t,\tau) {\mapsto} {\rm cov}[X(t),X(t+\tau)]}$ is almost periodic in t uniformly with respect to $\tau\in\mathbb{R}$. Then it admits a Fourier–Bohr decomposition: ${\rm cov}[X(t),X(t+\tau)] \sim \sum_{\lambda}a(\lambda,\tau) e^{{i}{\lambda}{t}}$. This paper deals with the estimation of the spectral covariance a(λ,τ) from a discrete time observation of the process $\{X(t){:}\ t\in\mathbb{R} \}$, when jitter and delay phenomena are present in conjunction with periodic sampling. Under mixing conditions, we establish the consistency and the asymptotic normality of empirical estimators as the sampling time step tends to 0 and the sampling period tends to infinity.