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Stochastic processes involving random deletion

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Mark D. Rothmann  and
Hammou El Barmi
Mark D. Rothmann
Affiliation:
University of Iowa
Hammou El Barmi*
Affiliation:
Kansas State University
*
∗∗ Postal address: Department of Statistics, Kansas State University, Manhattan, KS 66506, USA.
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Abstract

We consider a system where units having magnitudes arrive according to a nonhomogeneous Poisson process, remain there for a random period and then depart. Eventually, at any point in time only a portion of those units which have entered the system remain. Of interest are the finite time properties and limiting behaviors of the distribution of magnitudes among the units present in the system and among those which have departed from the system. We will derive limiting results for the empirical distribution of magnitudes among the active (departed) units. These results are also shown to extend to systems having stages or steps through which units must proceed. Examples are given to illustrate these results.

Keywords

nonhomogeneous Poisson processesrandom deletionweighted distributions

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60F15: Strong theorems 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 95 - 107
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

∗

Current address: Food and Drug Administration, 5600 Fishers Lane, Rockville, MD 20857-0001, USA. Email address: rothmannm@cder.fda.gov

References

Kalbfleisch, J. D., and Lawless, J. F. (1989). Inference based on retrospective ascertainment: an analysis of the data on transfusion-related AIDS. J Amer. Statist. Assoc. 84, 360–372.CrossRefGoogle Scholar
Resnick, S. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston.Google Scholar
Ross, S. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
Rothmann, M. D., and Russo, R. P. (1997). On the limiting proportion of types. J. Theoret. Prob. 10, 131–143.CrossRefGoogle Scholar
Vardi, Y. (1985). Empirical distributions in selection bias models. Ann. Statist. 13, 178–203.CrossRefGoogle Scholar