×

zbMATH — the first resource for mathematics

Systems of independent Markov chains. (English) Zbl 0657.60123
Consider an infinite system of independent Markov chains as a process on the space of particle configurations. The main results concern possible limiting distributions and equilibrium distribution for these processes. Necessary and sufficient conditions for this convergence are given.
If the underlying Markov chain is null recurrent, then every limiting distribution is a Cox process. In the positive recurrent case the limiting distribution is closely related to the invariant probability measure of the Markov chain. Also the transient case is treated.
Reviewer: U.Rösler

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barbour, A.D.; Hall, P., On the rate of Poisson convergence, Math. proc. camb. phil. soc., 95, 473-480, (1984) · Zbl 0544.60029
[2] Cox, J.T., Entrance laws for Markov chains, Ann. probab., 5, 533-549, (1977) · Zbl 0369.60079
[3] Cox, J.T.; Rosler, U., A duality relation for entrance and exit laws for Markov processes, Stochastic process. appl., 16, 141-156, (1983)
[4] Derman, C., Some contributions to the theory of denumerable Markov chains, Trans. amer. math. soc., 79, 541-555, (1955) · Zbl 0065.11405
[5] Dobrushin, R.L., On Poisson laws for distributions of particles in space, Ukrain. math. Z., 3, 127-134, (1956)
[6] Doob, J.L., Stochastic processes, (1953), Wiley New York · Zbl 0053.26802
[7] Lamperti, J., On null-recurrent Markov chains, Can. J. math., 12, 278-288, (1960) · Zbl 0133.10901
[8] Liemant, A.; Matthes, K., Verallgemeinerung eines satzes von dobrushin VI, Math. nachr., 80, 7-18, (1977) · Zbl 0383.60090
[9] Liggett, T.M., Random invariant measures for Markov chains, and independent particle systems, Z. wahrsch. verw. geb., 45, 297-313, (1978) · Zbl 0373.60076
[10] Matthes, K.; Kerstan, J.; Mecke, J., Infinitely divisible point processes, (1978), Wiley New York · Zbl 0383.60001
[11] Revuz, D., Markov chains, (1984), North-Holland Amsterdam · Zbl 0539.60073
[12] Stone, C., On a theorem of dobrushin, Ann. math. stat., 39, 1391-1401, (1968) · Zbl 0269.60045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.