Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.

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