# Markov chain, generalized

From Encyclopedia of Mathematics

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

[1] | J.L. Doob, "Stochastic processes" , Wiley (1953) |

#### Comments

#### References

[a1] | D. Freedman, "Markov chains" , Holden-Day (1975) |

[a2] | J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) |

[a3] | D. Revuz, "Markov chains" , North-Holland (1975) |

[a4] | V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) |

[a5] | E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) |

[a6] | A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , 1–2 , Gordon & Breach (1965–1968) (Translated from French) |

**How to Cite This Entry:**

Markov chain, generalized.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_generalized&oldid=13364

This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article