# Markov chain, class of positive states of a

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

of satisfy

for any , , , and

where is the return time to the state :

for a discrete-time Markov chain, and

for a continuous-time Markov chain. When , is called a zero class of states (class of zero states).

States in the same positive class have a number of common properties. For example, in the case of discrete time, for any the limit relation

holds; if

is the period of state , then for any and is called the period of the class ; for any the limit relation

holds. A discrete-time Markov chain such that all its states form a single positive class of period 1 serves as an example of an ergodic Markov chain (cf. Markov chain, ergodic).

#### References

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

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

#### Comments

Cf. also Markov chain, class of zero states of a for additional refences.

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Markov chain, class of positive states of a.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_class_of_positive_states_of_a&oldid=14075