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On the superposition of m-dimensional point processes

Published online by Cambridge University Press:  14 July 2016

Erhan Çinlar
Erhan Çinlar*
Affiliation:
Northwestern University, Evanston, Illinois
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Abstract

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 1 , April 1968 , pp. 169 - 176
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

Cox, D. R. and Smith, W. L. (1954) On the superposition of renewal processes. Biometrika 41, 91–99.Google Scholar
Goldman, J. R. (1967) Stochastic point processes: limit theorems. Ann. Math. Statist. 38, 771–779.Google Scholar
Khintchine, A. I. (1955) Mathematical Methods in the Theory of Queueing. Translation (1960), Griffin, London.Google Scholar
Neuts, M. F. (1966) The single server queue with Poisson input and semi-Markov service times. J. Appl. Prob. 3, 202–230.CrossRefGoogle Scholar
Palm, C. (1943) Intensitätsschwankungen im Fernsprechverkehr. Ericsson Techniks 44, 1–189.Google Scholar
Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.Google Scholar
Pyke, R. and Shaufele, R. (1966) The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Statist. 37, 1439–1462.Google Scholar
Ten Hoopen, M. and Reuver, H. A. (1966) The superposition of random sequences of events. Biometrika 53, 383–389.CrossRefGoogle Scholar
Welch, P. D. (1965) On the busy period of a facility which serves customers of several types. J.R. Statist. Soc., B 27, 361–370.Google Scholar