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.
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