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On pair and tuple formation under independent Poisson or renewal arrival processes

Part of: Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

K. Borovkov
K. Borovkov*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia. Email address: k.borovkov@ms.unimelb.edu.au
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Abstract

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

Keywords

Pair formationPoisson processrenewal process

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60F99: None of the above, but in this section 60J75: Jump processes 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1138 - 1144
Copyright
Copyright © Applied Probability Trust 2004 

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References

Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Neuts, M. F., and Alfa, A. S. (2004). Pair formation in a Markovian arrival process with two event labels. J. Appl. Prob. 41, 1124–1137.CrossRefGoogle Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford University Press.Google Scholar
Siegmund, D. (1985). Sequential Analysis. Springer, New York.Google Scholar