Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T04:35:29.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

Published online by Cambridge University Press:  14 July 2016

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

Abstract

The asymptotic behaviour of the sequence (𝒞n(ω), wc,n(ω)/n), is studied where 𝒞n(ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n) until time n and wc,n(ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ωc) which represents uniquely the chain (ξ n) as a circuit chain, and ω c is given a probabilistic interpretation.

Keywords

CIRCUIT CHAINSMULTIPLE MARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 3 , September 1990 , pp. 545 - 556
Copyright
Copyright © Applied Probability Trust 1990 

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.)

References

[1] Chung, K. L. (1967) Markov Chains with Stationary Transition Probability. Springer-Verlag, New York.Google Scholar
[2] Harris, T. (1952) First passage and recurrence distributions. Trans. Amer. Math. Soc. 73, 471–486.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. (1973) On multiple Markovian dependence. Proc. Fourth Conf. Probability Theory, Brasov, 1971, Publishing House of the Romanian Academy, Bucharest, 65–71.Google Scholar
[5] Kalpazidou, S. (1989) On multiple circuit chains with a countable infinity of states. Stoch. Proc. Appl. 31, 51–70.Google Scholar
[6] Kalpazidou, S. (1989) Representation of denumerable Markov chains with multiple states by weighted circuits. J. Appl. Prob. 26, 23–35.Google Scholar
[7] Kalpazidou, S. (1988) On circuit chains defined by forward and backward passages. Stoch. Anal. Appl. 6, 397–416.CrossRefGoogle Scholar
[8] Minping, Qian and Min, Qian (1979) The decomposition into a detailed balance part and a circulation part of an irreversible stationary Markov chain. Scientia Sinica Special Issue II, 69–79.Google Scholar
[9] Minping, Qian, Min, Qian and Cheng, Qian (1982) Circulation distribution of a Markov chain. Scientia Sinica A 25, 31–40.Google Scholar