mc_lim - Winter 2010 Pstat160a Hand-out MC#5 Limiting...

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Winter 2010 Pstat160a Hand-out MC#5 Limiting probabilities 1. Examples from handout MC #2 a) Second matrix there: P = . 3 . 6 0 0 . 1 0 . 4 . 6 0 0 0 . 6 . 4 0 0 0 0 0 . 4 . 6 0 0 0 1 0 classes: { 1 } : T { 2,3 } :R, d=1; { 4,5 } : R ,d = 1 P 200 = 0 . 4286 . 4286 . 0893 . 0536 0 . 5 . 5 0 0 0 . 5 . 5 0 0 0 0 0 . 625 . 375 0 0 0 . 625 . 375 b) Fourth matrix there: P = . 3 . 6 0 0 . 1 0 . 4 . 6 0 0 . 4 . 6 0 0 0 0 0 . 4 0 . 6 0 0 0 1 0 classes: irreducible, aperiodic ( d = 1) P 100 = . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 P 113 = . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 . 1667 . 4583 . 2917 . 0417 . 0417 2. Limiting and stationary probabilities Let X n be a Markov chain with states S (say S = 1 , 2 , ··· ,N ). The limiting probability , (if it exists) the the limit lim n →∞ P ( X n = j | X 0 = i ). Note that this is exactly what is happening with the matrix P 100 in b) above. The stationary probability (if it exists) for X n is a probability (row) vector π = { π i , i ∈ S} such that if P ( X n = i ) = π i then P ( X n +1 = i ) = π i as well. We sometimes use the notation π ( j ) for π j . Note that by definition of stationarity and conditioning on the previous state, we see that π needs to satisfies π j = P ( X n +1 = j ) = X i ∈S P ( X n +1 = j
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mc_lim - Winter 2010 Pstat160a Hand-out MC#5 Limiting...

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