Random sampling of random processes: Stationary point processes

Abstract

This is the first of a series of papers treating randomly sampled random processes. Spectral analysis of the resulting samples pre-supposes knowledge of the statistics of tn, the random point process whose variates represent the sampling times. We introduce a class of stationary point processes, whose stationarity (as characterized by any of several equivalent criteria) leads to wide-sense stationary sampling trains when applied to wide-sense stationary processes. Of greatest importance are the nth forward [backward] recurrence times (distances from t to the nth point thereafter [preceding]), whose distribution functions prove more useful to the computation of covariances than interval statistics, and which possess remarkable properties that facilitate the analysis. The moments of the number of points in an interval are evaluated by weighted sums of recurrence time distribution functions, the moments being finite if and only if the associated sum converges. If the first moment is finite, these distribution functions are absolutely continuous, and obey some convexity relations. Certain formulas relate recurrence statistics to interval length statistics, and conversely; further, the latter are also suitable for a direct evaluation of moments of points in intervals. Our point process requires neither independent nor identically distributed interval lengths. It embraces most of the common sampling schemes (e.g., periodic, Poisson, jittered), as well as some new models. Of particular interest are point processes obtained from others by a random deletion of points (skip processes), as for instance a jittered cyclically periodic process with (random or systematic) skipping. Computation of the statistics for several point processes yields new results of interest not only for their own sake, but also of use for spectral analyses appearing in other papers of this series.