# Markov chain, class of zero states of a

From Encyclopedia of Mathematics

A set of states of a homogeneous Markov chain with state space such that

for any ,

for any , , , and

(*) |

for any , where is the return time to the state :

for a discrete-time Markov chain, and

for a continuous-time Markov chain.

As in the case of a class of positive states (in the definition of a positive class (*) is replaced by ), states belonging to the same zero class have a number of common properties. For example, for any states of a zero class ,

An example of a Markov chain whose states form a single zero class is the symmetric random walk on the integers:

where are independent random variables,

#### References

[1] | K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967) |

#### Comments

Cf. also Markov chain, class of positive states of a.

#### References

[a1] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966) |

[a2] | D. Freedman, "Markov chains" , Holden-Day (1975) |

[a3] | M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980) |

[a4] | J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) |

[a5] | J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976) |

[a6] | D. Revuz, "Markov chains" , North-Holland (1975) |

[a7] | V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) |

[a8] | E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) |

**How to Cite This Entry:**

Markov chain, class of zero states of a.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_class_of_zero_states_of_a&oldid=18811

This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article