Arrival times of Cox process with independent increment with application to prediction problems

Abstract: Properties of arrival times are studied for a Cox process with independent (and stationary) increments. Under a reasonable setting the directing random measure is shown to take over independent (and stationary) increments of the process, from which the sets of arrival times and their numbers in disjoint intervals are proved to be independent (and stationary). Moreover, we derive the exact joint distribution of these quantities with Gamma random measure, whereas for a general random measure the method of calculation is presented. Based on the derived properties we consider prediction problems for the shot noise process with Cox process arrival times which trigger additive processes off. We obtain a numerically tractable expression for the predictor which works reasonably in view of numerical experiments.