Winter 2010Pstat160aHand-out MC#3Classification of states1. Communication classes, irreducibility:2 statesi, jcommunicate if the process can go fromitojand fromjtoiin one or more steps.The set of statesthat communicate with each others are in the same communication class.The chain is irreducible if there is only onecommunication class.2. Transience/recurence:A stateiis recurrent if the probability to come back toi(so starting fromi) 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 statesare recurrent (or transient).3. Period:The period of a stateiis the greatest common divisordofnsuch thatP(n)ii>0. That isd≥2 ifP(n)ii>0 on ifn=kdfor anyklarge enough. So ifPii>0, d= 1. Periodicity is also a class property.4. Examples:P=
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