 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ib/paper-1/11h/b/solution

i

satisfies

Pr_{i} (T_{i}^{+} < \infty) = 1

, where

T_{i}^{+} = in f {n \geq 1 : X_{n} = i}

. Saying that the transition matrix is recurrent means that every state is recurrent. In an irreducible Markov chain it suffices that one state is recurrent, since recurrence is a communicating-class property.

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