Basic Stochastic Processes by Tomasz Zastawniak & Zdzisław Brzeźniak

Author:Tomasz Zastawniak & Zdzisław Brzeźniak
Language: eng
Format: epub
Publisher: Springer-Verlag Wien 2012
Published: 2014-06-16T16:00:00+00:00

Definition 5.9

Suppose that ξn, n ∈ ℕ, is a Markov chain on a countable state space S. 1) A set C ⊂ S is called closed if once the chain enters C it will never leave it, i.e. (5.45)

2) A set C ⊂ S is called irreducible if any two elements i, j of C intercommunicate, i.e. for all i, j ∈ C there exists an n ∈ ℕ such that pn(j∣i) > 0.

Theorem 5.3

Suppose that ξn, n ∈ ℕ, is a Markov chain on a countable state space S. Then (5.46)

where T is the set of all transient states in S and each Cj is a closed irreducible set of recurrent states.

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