Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-18T20:35:06.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite non-homogeneous Markov chains: Asymptotic behaviour

Published online by Cambridge University Press:  01 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

Keywords

STOCHASTIC MATRICESNON-HOMOGENEOUS MARKOV CHAINTAIL σ-FIELDATOMIC SETMARTINGALESσ-FIELD COMPLETIONWEAK ERGODICITY
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 3 , September 1976 , pp. 502 - 516
Copyright
Copyright © Applied Probability Trust 1976 

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

Blackwell, D. (1945) Finite nonhomogeneous Markov chains. Ann. Math. 46, 594–599.CrossRefGoogle Scholar
Blackwell, D. and Freedman, D. (1964) The tail σ-field of a Markov chain and a theorem of Orey. Ann. Math. Statist. 38, 1921–1925.Google Scholar
Cohn, H. (1970) On the tail σ-algebra of the finite inhomogeneous Markov chains. Ann. Math. Statist. 41, 2175–2176.Google Scholar
Cohn, H. (1972) On the Borel–Cantelli lemma. Israel J. Math. 12, 11–16.Google Scholar
Cohn, H. (1974a) On the tail events of a Markov chain. Z. Wahrscheinlichkeitsth. 29, 65–72.CrossRefGoogle Scholar
Cohn, H. (1974b) A ratio limit theorem for the finite nonhomogeneous Markov chains. Israel J. Math. 19, 329–334.CrossRefGoogle Scholar
Doeblin, W. (1937) Le cas discontinu des probabilités en chaîne. Publ. Fac. Sci. Univ. Masaryk (Brno), 236.Google Scholar
Hajnal, J. (1958) Weak ergodicity in non-homogeneous Markov chains. Proc. Camb. Phil. Soc. 54, 233–246.Google Scholar
Iosifescu, M. (1972a) On two recent papers on ergodicity in non-homogeneous Markov chains. Ann. Math. Statist. 43, 1732–1736.Google Scholar
Iosifescu, M. (1972b) On finite tail σ-algebras. Z. Wahrscheinlichkeitsth. 24, 159–166.Google Scholar
Kingman, J. F. C. (1975) Geometrical aspects of the theory of non-homogeneous Markov chains. Math. Proc. Camb. Phil. Soc. 77, 171–185.Google Scholar
Kolmogorov, A. (1936) Zur Theorie der Markoffschen Ketten. Math. Annal. 112, 155–160.CrossRefGoogle Scholar
Loève, M. (1965) Probability Theory, 3rd edn. Van Nostrand, Princeton.Google Scholar
Lopez, A. (1961) Problems in Stable Population Theory. Office of Population Research, Princeton.Google Scholar
Moran, P. A. P. (1962) Polysomic inheritance and the theory of shuffling. Sankhyā A 24, 63–72.Google Scholar
Senchenko, D. V. (1972) The final σ-algebra of an inhomogeneous Markov chain with a finite number of states. Math. Notes 12, 610–613.Google Scholar
Seneta, E. (1973a) On the historical development of the theory of finite non-homogeneous Markov chains. Proc. Camb. Phil. Soc. 74, 507–513.CrossRefGoogle Scholar
Seneta, E. (1973b) Non-Negative Matrices. Allen and Unwin, London.Google Scholar
Wolfowitz, J. (1963) Products of indecomposable, aperiodic, stochastic matrices. Proc. Amer. Math. Soc. 14, 733–737.CrossRefGoogle Scholar