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Unformatted text preview: Winter 2010 Pstat160a Handout MC#3 Classiﬁcation of states 1. Communication classes, irreducibility: 2 states i, j communicate if the process can go from i to j and from j to i in one or more steps. The set of states that communicate with each others are in the same communication class. The chain is irreducible if there is only one communication class. 2. Transience/recurence: A state i is recurrent if the probability to come back to i (so starting from i) is 1, it is transient if this probability is < 1. Transience and recurrence is a communication class property, so we say that a class is recurrent (or transient) if its states are recurrent (or transient). 3. Period: The period of a state i is the greatest common divisor d of n such that Pii
(n) > 0. That is d ≥ 2 if Pii ( n) > 0 on if n = kd for any k large enough. So if Pii > 0, d = 1. Periodicity is also a class property. 4. Examples: .3 0 P = .4 0 0 .6 .4 .6 0 0 0 .6 0 0 0 0 0 0 .4 1 .1 0 0 .6 0 classes: .3 .6 .4 .6 0 0 0 .6 .4 0 0 0 0 0 .4 1 0 P = 0 0 0 .1 0 0 .6 0 classes: .3 0 P = 0 0 0 .6 .4 .6 0 0 0 .1 .6 0 0 .4 0 0 .4 0 .6 010 0 classes: .3 .6 0 .6 0 .4 0 0 .4 P = .4 .6 0 0 00 0 .1 0 0 0 0 0 .6 1 0 classes: .3 0 P = .4 0 0 .6 .4 .6 0 0 0 .1 .6 0 0 0 0 0 .4 0 .6 0 0 0 1 classes: ...
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This note was uploaded on 01/10/2011 for the course STAT 160A taught by Professor Bonnet during the Winter '10 term at UCSB.
 Winter '10
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