Random Sampling of Random Processes: Stationary Point Processes

This is the first of a series of papers treating randomly sampled random processes. Spectral analysis of the resulting samples presupposes knowledge of the statistics of 1 t~}, the random point process whose variates represent the sampling times. We introduce a class of s ta t ionary point processes, whose s ta t ionar i ty (as characterized by any of several equivalent criteria) leads to wide-sense s tat ionary sampling trains when applied to wide-sense s tat ionary processes. Of greatest importance are the nth forward [backward] recurrence times (distances from t to the nth point thereafter [preceding!), whose dist r ibut ion functions prove more useful to the computat ion of covariances than interval statistics, and which possess remarkable properties that facili tate the analysis. The moments of the number of points in an interval are evaluated by weighted sums of recurrence t ime distr ibution 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 la t te r are also suitable for a direct evaluation of moments of points in intervals. Our point process requires neither independent nor identical ly distr ibuted interval lengths. I t embraces most of the common sampling schemes (e.g., periodic, Poisson, i i t tered) , as well as some new models. Of part icular interest are point processes obtained from others by a random deletion of points (skip processes), as for instance a j i t te red cyclically periodic process with (random or sys-