MATH 303 Quiz: HW2
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Hint: use the discrete fourier transform to diagonalize the transition matrix. That is, express the transition matrix in the orthonor- mal basis e(0), e(1), . , e(n 1) of cn which has the following components in the standard basis: e(k) , n 1, i = 1). j = 1. 2 i e: consider an irreducible markov chain with nite state space s = {0, 1, . , n}. (a) starting at state i, what is the probability that it will ever visit state j? (i, j arbi- trary). (b) suppose that pj j pij = i for all i. Let xi = p(visit n before 0 | start at i). Hint: derive a system of linear equations that the xi satisfy, and show. N solves these equations. that xi = i that xi = i. Recommended problems: these provide additional practice but are not to be handed in.
