Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T09:48:30.755Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric ρ-Mixing Property of the Interarrival Times of a Stationary Markovian Arrival Process

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Loïc Hervé  and
James Ledoux
Loïc Hervé*
Affiliation:
INSA de Rennes, IRMAR CNRS UMR 6625
James Ledoux*
Affiliation:
INSA de Rennes, IRMAR CNRS UMR 6625
*
∗ Postal address: INSA, 20 avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
∗ Postal address: INSA, 20 avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note, the sequence of the interarrivals of a stationary Markovian arrival process is shown to be ρ-mixing with a geometric rate of convergence when the driving process is ρ-mixing. This provides an answer to an issue raised in the recent work of Ramirez-Cobo and Carrizosa (2012) on the geometric convergence of the autocorrelation function of the stationary Markovian arrival process.

Keywords

Markov renewal process

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 598 - 601
Copyright
© Applied Probability Trust 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Bradley, R. C. (2005). Introduction to strong mixing conditions (volume I). Tech. Rep., Indiana University.Google Scholar
Ferré, D., Hervé, L. and Ledoux, J. (2012). Limit theorems for stationary Markov processes with L 2-spectral gap. Ann. Inst. H. Poincaré Prob. Statist. 48, 396–423.Google Scholar
Ramirez-Cobo, P. and Carrizosa, E. (2012). A note on the dependence structure of the two-state Markovian arrival process. J. Appl. Prob. 49, 295–302.CrossRefGoogle Scholar
Ramirez-Cobo, P., Lillo, R. E. and M, Wiper.. (2010). Nonidentiability of the two-state Markovian arrival process. J. Appl. Prob. 47, 630–649.Google Scholar
Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Springer, New York.Google Scholar