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March 16th, 2009, 04:41 PM   #1
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Markov chain and classifying states

Given the transition matrices below, find all communicating classes and classify all states:
The probabilities don't matter for this exercise, so I'm just going to put ones for non-zero entries, even though I know this is absurd mathematically:
(A)
1 1 0 1
1 0 0 1
1 0 0 0
1 0 1 0
I believe that the communicating classes are {1,2,4} and {3} and all these are ergodic.

(B)
1 1 1 1
0 0 1 1
1 0 1 0
0 0 1 0
I think the communicating classes are {1,3}, {2}, and {4}.
I'm not quite sure, but I believe states 1,3, and 4 are ergodic, and 3 is of period 2.

(C)
1 1 0 0
1 1 0 0
0 0 0 1
0 0 1 1
The communicating classes are {1,2} and {3,4} with {1,2} of period 2 and {3,4} ergodic.

Is this right? I think I have the communicating classes right, but I'm not sure about the state classifications. Thanks in advance for any help.
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