  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.
