By John G. Kemeny, J. Laurie Snell
Finite Markov Chains: With a brand new Appendix "Generalization of a basic Matrix"
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Extra info for Finite Markov Chains: With a New Appendix ''Generalization of a Fundamental Matrix''
Sample text
P(n). In the case of a Markov chain process, all the Tknys are the same and we obtain the following fundamental theorem. 2 THEOREM. Let Tn be the induced measure for the outcome function fn for a finite Markov chain with initial probability vector 77 0 and transition matrix P. Then 7771 = Tro• P n • This theorem shows that the key to the study of the induced measures for the outcome functions of a finite Markov chain is the study of the powers of the transition matrix. The entries of these powers have themselves an interesting probabilistic interpretation.
II described are u 1 , u2, , ux, then our matrix will appear as follows (where k is taken as 5, for the sake ofillustration) : u2: PR21 P2 P3 u3: u4: 0 R4 u5: P4 R5 P5 Here the P i represent transition matrices within a given equivalence class. The region 0 consists entirely of 0's. The matrix R i will be entirely 0 if P i is an ergodic set, but will have non-zero elements otherwise. In this form it is easy to see what happens as P is raised to powers. Each power will be a matrix of the same form ; in pm we still have zeros in the upper region, and we simply have Pni in the diagonal regions.
This corollary is useful if we are interested in a single absorbing state. 9 THEOREM. If B* is the r x r matrix whose entry b*il gives the probability of being absorbed in Sy, starting in si, for all states s i and s i , then PB* = B*. PROOF. If sjE T, then b*ii= 0. Hence the last s columns of B* are 0. Consider s, absorbing. 7. If si is also absorbing, then b*,;=d i f. NR=R+(N-1)R----NR=B.
