Bootstrap for the second-order analysis of Poisson-sampled almost periodic processes

Abstract

In this paper we consider a continuous almost periodically correlated process $\{X(t),t\in\mathbb{R}\}$ that is observed at the jump moments of a stationary Poisson point process $\{N(t),t\geq0\}$. The processes $\{X(t),t\in\mathbb{R}\}$ and $\{N(t),t\geq0\}$ are assumed to be independent. We define the kernel estimators of the Fourier coefficients of the autocovariance function of $X(t)$ and investigate their asymptotic properties. Moreover, we propose a bootstrap method that provides consistent pointwise and simultaneous confidence intervals for the considered coefficients. Finally, to illustrate our results we provide a simulated data example.