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Markov chain, generalized

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2010 Mathematics Subject Classification: Primary: 60J10 [MSN][ZBL]

A sequence of random variables with the properties:

1) the set of values of each is finite or countable;

2) for any and any ,

(*)

A generalized Markov chain satisfying (*) is called -generalized. For , (*) is the usual Markov property. The study of -generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables whose values are in one-to-one correspondence with the values of the vector

The sequence forms an ordinary Markov chain.

References

[D] J.L. Doob, "Stochastic processes" , Wiley (1953) MR1570654 MR0058896 Zbl 0053.26802

Comments

References

[F] D. Freedman, "Markov chains" , Holden-Day (1975) MR0686269 MR0681291 MR0556418 MR0428472 MR0292176 MR0237001 MR0211464 MR0164375 MR0158435 MR0152015 Zbl 0501.60071 Zbl 0501.60069 Zbl 0426.60064 Zbl 0325.60059 Zbl 0322.60057 Zbl 0212.49801 Zbl 0129.30605
[KS] J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704
[Re] D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045
[Ro] V.I. Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002
[S] E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) MR2209438 Zbl 0471.60001
[BF] A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , 1–2 , Gordon & Breach (1965–1968) (Translated from French) Zbl 0185.44502 Zbl 0159.45802
How to Cite This Entry:
Markov chain, generalized. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Markov_chain,_generalized&oldid=26566
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article