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the probability that the chain occupies each subset of A in nitely often over unbounded time is 1 More informally, when a chain moves into a recurrent state, it stays there forever and visits every substate in nitely often A recurrent Markov chain is positive recurrent if the mean time to return to A is bounded; otherwise, it is called null recurrent (Doeblin 1940) Unfortunately, given unbounded and continuous state spaces, we have to work with a slightly more complicated version of recurrence If there exists a - nite probability measure P on the measure space H such that an -irreducible Markov chain, n , at time n has the property P ( n A) = 1, A H where P > 0, it is Harris recurrent (Harris 1956; Athreya and Ney 1978).

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This is necessary because in the continuous case an aperiodic, -irreducible chain with an invariant distribution on an unbounded continuous state space that is not Harris recurrent has a positive probability of getting stuck forever in an area bounded away from convergence, given a starting point there So the purpose of the Harris de nition is to avoid worrying about the existence of a pathological null set in Rk Fortunately, all of the standard MCMC algorithms implemented on (naturally) nite-state computers are Harris recurrent (Tierney 1994) From this point on we will use recurrent generically to mean the simpler de nition of recurrent for closed or discrete sets and Harris recurrent for unbounded continuous sets 532.

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5 Transience First de ne for a set A the expected number of visits by chain n to A in the limit A = I( n A) , which is just a summed indicator function that counts hits n=1 Both transience and recurrence can be de ned in terms of the expectation for A : The set A is uniformly transient if M < E[ A ] M A A single state in the discrete state space case is transient if E[ ] < The set A is recurrent if E[ A ] = A A single state in the discrete state space case is recurrent if: E[ ] = ..

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