Determining the power of the transition matrix

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1. Define (r) to be the smallest integer such that for all regular Markov chains with states, the nth power of the transition matrix has all entries positive. It has been shown,14 that (r) =r2+ 2.

(a) Define the transition matrix of an r-state Markov chain as follows:  For states si, with i = 1, 2, . . . , r-2,P(i, i+1) = 1, P(r-1, r) = P(r-1, 1) = 1/2, and P(r, 1) = 1.  Show that this is a regular Markov  chain.

(b) For = 3, verify that the fifth power is the first power that has no zeros.

(c) Show that, for general r, the smallest such that Phas all entries positive is (r).

Reference no: EM13985132

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