Journal of Statistical Planning and Inference 100 (2002) 171–183 www.elsevier.com/locate/jspi Filtration of ASTA: a weak convergence approach Muhammad El-Taha a ; ∗ , Shaler Stidham Jr. b a Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME 04104-9300, USA b Department of Operations Research, CB 3180, Smith Building, University of North Carolina, Chapel Hill, NC 27599-3180, USA Abstract Consider a stochastic process (X;A), where X represents the evolution of a system over time, and A is an associated point process that has stationary independent increments. Suppose we are interested in estimating the time average frequency of the process X being in a set of states. Often it is more convenient to have a sampling procedure for estimating the time average based on averaging the observed values of X (Tn)(Tn being a point of A) over a long period of time: the event average of the process. In this paper we examine the situation when the two procedures—event averaging and time averaging—produce the same estimate (the ASTA property: Arrivals See Time Averages). We prove a result stronger than ASTA. Under a lack-of-anticipation assumption we prove that the point process, A, restricted to any set of states, has 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. We develop our results in the more general setting of a stochastic process (X;A), that is, a process with an imbedded cumulative process, A = {A(t );t ¿ 0}, which is assumed to be a Levy process with non-decreasing sample paths. This framework allows for modeling uid processes, as well as compound Poisson processes with non-integer increments. First, we state the result in discrete time; the discrete-time result is then extended to the continuous-time case using limiting arguments and weak-convergence theory. As a corollary we give a proof of ASTA under weak conditions and a simple, intuitive proof of (Poisson Arrivals See Time Averages) under the standard conditions. The results are useful in queueing and statistical sampling theory. c  2002 Elsevier Science B.V. All rights reserved. Keywords: ASTA; PASTA; Queueing; Weak convergence; Time average; Event average; Cumulative processes; Point processes; Fluid models 1. Introduction Consider a queueing process with an associated arrival process satisfying a lack-of- anticipation assumption (LAA), which says that future arrivals occur independently of the present and past behavior of the queueing process. If the arrival process is * Corresponding author. E-mail addresses: eltaha@usm.maine.edu (M. El-Taha), sandy@stidham.or.unc.edu (S. Stidham Jr.). 0378-3758/02/$ - see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0378-3758(01)00131-8