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mc2 - Winter 2010 Pstat160a Hand-out MC#3 Classication of...

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Winter 2010 Pstat160a Hand-out 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 P ( n ) ii > 0. That is d 2 if P ( n ) ii > 0 on if n = kd for any k large enough. So if P ii > 0 , d = 1. Periodicity is also a class property. 4. Examples: P =
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