A filtered ASTA property

Abstract

Recently, the PASTA (Poisson Arrivals See Time Averages) property has been extended to ASTA (Arrivals See Time Averages) by eliminating the need for Poisson arrivals and weakening the LAA (Lack of Anticipation Assumption). This paper presents a strengthening of ASTA under the original LAA of Wolff. We consider a stochastic processX with an associated point processN that admits a stochastic intensity and satisfies LAA. Various authors have noted in various contexts that ASTA holds if and only if the arrival intensity is state independent. For a class of point processes that includes doubly stochastic as well as ordinary Poisson processes, we prove that the point process obtained by restricting the processX to any given set of states not only has the same intensity but also the same probabilistic structure as the original point process. In particular, if the original point process is Poisson, the new point process is still Poisson with the same parameter as the original point process. For a discrete-time version, of interest in its own right, we provide a simple proof of a strengthened version of ASTA in discrete time. Unlike other discrete-time versions of ASTA, ours is valid for point processes with stationary but not necessarily independent increments. The continuous-time results are obtained using martingale theory. A corollary is a simple proof of PASTA under conditions that require only that the relevant limits exist. Our results may also provide some insight into characterizing Poisson flows in queueing systems.