Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T01:21:51.761Z Has data issue: false hasContentIssue false

Exact aggregation of absorbing Markov processes using the quasi-stationary distribution

Published online by Cambridge University Press:  14 July 2016

James Ledoux
Affiliation:
IRISA-INRIA
Gerardo Rubino
Affiliation:
IRISA-INRIA
Bruno Sericola*
Affiliation:
IRISA-INRIA
*
Postal address for all authors: IRISA-INRIA, Campus de Beaulieu 35042 Rennes, France.

Abstract

We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was partially supported by the Regional Council of Britanny under Grant 290C2010031305061.

References

[1] Buiculescu, (1972) Quasi-stationary distributions on continuous Markov chains. Rev. Roumaine Math. Pures Appl. 18(7), 10131023.Google Scholar
[2] Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discretetime finite Markov chains. J. Appl. Prob. 2, 88100.CrossRefGoogle Scholar
[3] Kemeny, J. G. and Snell, J. L. (1976) Finite Markov Chains. Springer-Verlag, New York.Google Scholar
[4] Ledoux, J., Rubino, G. and Sericola, B. (1992) Agrégation faible des processus de Markov absorbants. Technical Report No. 1736, INRIA, 1992. Campus de Beaulieu, 35042 Rennes Cedex, France.Google Scholar
[5] Matheiss, T. H. and Rubin, D. S. (1980) A survey and comparison of methods for finding all vertices of convex polyhedral sets. Math. Oper. Res. 5, 167185.CrossRefGoogle Scholar
[6] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[7] Rubino, G. and Sericola, B. (1991) A finite characterization of weak lumpable Markov processes. Part I: The discrete time case. Stoch. Proc. Appl. 38, 195204.CrossRefGoogle Scholar
[8] Rubino, G. and Sericola, B. (1993) A finite characterization of weak lumpable Markov processes Part II: The continuous time case. Stoch. Proc. Appl. 45, 115125.CrossRefGoogle Scholar
[9] Seneta, E. Non-negative Matrices and Markov Chains. Springer-Verlag, New York.CrossRefGoogle Scholar