Lyapunov functions. Let Xn be an irreducible Markov chain with state space f0; 1; 2; : :

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Lyapunov functions. Let Xn be an irreducible Markov chain with state space f0; 1; 2; : : :g and let   0 be a function with limx!1 .x/D1, and Ex.X1/  .x/ when x  K. Then Xn is recurrent. This abstract result is often useful for proving recurrence in many chains that come up in applications and in many cases it is enough to consider .x/ D x.

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