STAT 333 markov_renewal

# STAT 333 markov_renewal - Classifying States of a Markov...

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Unformatted text preview: Classifying States of a Markov Chain Definitions : 1. Accessible ( j → k )- State k is accessible from state j if it is possible to find a path from state j to k . That is, P ( n ) jk > for some n ≥ . 2. Communicate ( j ↔ k )- States j and k communicate iff state k is accessible from j and vice-versa. That is, j → k and k → j . Additionally, communication is an equivalence relation since • j ↔ j (reflexive) • j ↔ k implies k ↔ j (symmetric) • if i ↔ j and j ↔ k , then i ↔ k (transitive) 3. We can partition S (the set of states of the MC) into equivalence/communicating classes. Assign states i and j to the same class if and only if each of these states communicate with each other. There are two types of classes: • Closed Class: A communicating class C is closed iff i ∈ C and i → j implies that j ∈ C . Therefore, P ij = 0 for all i ∈ C and j / ∈ C . • Open Class: A communicating class C is open iff for some i ∈ C , there exists j / ∈ C such that...
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## This note was uploaded on 01/21/2012 for the course STAT 333 taught by Professor Chisholm during the Fall '08 term at Waterloo.

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STAT 333 markov_renewal - Classifying States of a Markov...

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