Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-04-25T05:41:37.225Z Has data issue: false hasContentIssue false
  • English
  • Français

On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case

Part of: Markov processes Circuits, networks

Published online by Cambridge University Press:  14 July 2016

S. Kalpazidou
S. Kalpazidou*
Affiliation:
Aristotle University of Thessaloniki
*
∗ Postal address: Department of Mathematics, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.

Keywords

PROBABILISTIC CIRCUIT REPRESENTATION OF MARKOV PROCESSESDETERMINISTIC CIRCUIT REPRESENTATION OF MARKOV PROCESSES

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 94C15: Applications of graph theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 28 - 39
Copyright
Copyright © Applied Probability Trust 1993 

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.)

Article purchase

Temporarily unavailable

References

[1] Chung, K. L. (1967) Markov chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, Berlin.Google Scholar
[2] Chung, K. L. (1988) Reminiscences of some of Paul Levy's ideas in Brownian motion and in Markov chains. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque 157–158, 29-36.Google Scholar
[3] Iosifescu, M. (1969) Sur les chaînes de Markov multiples. Bull. Inst. Internat. Statist. 43(2), 333–335.Google Scholar
[4] Iosifescu, M. and Tautu, P. (1973) Stochastic Processes and Applications in Biology and Medicine, I, Theory. Springer-Verlag, Berlin. Edit. Acad., Bucharest.Google Scholar
[5] Kalpazidou, S. (1988) On the representation of finite multiple Markov chains by weighted circuits. J. Multivariate Anal. 25, 241–271.CrossRefGoogle Scholar
[6] Kalpazidou, S. (1990) Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains. J. Appl. Prob. 27, 545–556.CrossRefGoogle Scholar
[7] Kalpazidou, S. (1991) Continuous parameter circuit processes with finite state space. Stoch. Proc. Appl. 39, 301–323.CrossRefGoogle Scholar
[8] Kalpazidou, S. (1992) On the weak convergence of sequences of circuit processes: a probabilistic approach. J. Appl. Prob. 29, 374–383.CrossRefGoogle Scholar
[9] Kalpazidou, S. (1993) On the weak convergence of sequences of circuit processes: a deterministic approach.Google Scholar
[10] Levy, P. (1951) Systèmes markoviens et stationnaires. Cas dénombrable. Ann. Sci. Ecole Norm. Sup. 68, 327–381.CrossRefGoogle Scholar
[11] Levy, P. (1958) Processus markoviens et stationnaires. Cas dénombrable. Ann. Inst. H. Poincaré 16, 7–25.Google Scholar
[12] Minping, Qian, Min, Qian and Cheng, Qian (1982) Circulation distribution of a Markov chain. Scientia Sinica A25, 31–40.Google Scholar