SOME CHARACTERIZATIONS OF THE POISSON-PROCESS AND GEOMETRIC RENEWAL PROCESS

Let gamma(t) and delta(t) denote the residual life at t and current life at t, respectively, of a renewal process {A(t), t greater-than-or-equal-to 0), with {X(n), n greater-than-or-equal-to 1} the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as E(G(delta(t))\A(t) = n) = E(G(X1)\A(t) = n), for-all t greater-than-or-equal-to 0, holds for a single positive integer n, then {A(t), t greater-than-or-equal-to 0} is a Poisson process. On the other hand, for a delayed renewal process {A(D)(t), t greater-than-or-equal-to 0} with gamma(t)D, the residual life at t, we find that for some fixed positive integer n, if E(G(gamma(t)D)\A(D)(t) = n) is independent of t, then {A(D)(t), t greater-than-or-equal-to 0} is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of gamma(t) and delta(t).