STOCHASTIC ORDERING AND THINNING OF POINT-PROCESSES

We study the stochastic ordering of random measures and point processes generated by a partial order mu < nu if mu(B) less-than-or-equal-to nu(B) for all bounded Borel subsets B of the state space. For two stochastically ordered simple point processes on (0, infinity) a condition is derived that the former can be realized as a thinning of the latter. The condition is expressed by the stochastic intensity function. The results are applied to renewal processes and Markov renewal processes, in particular to Poisson processes. For a renewal process N with a decreasing failure rate it is shown that {N(.+t), t greater-than-or-equal-to 0} is an isotonically decreasing family of point processes.