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In other words, jj is the expected number of transitions needed to return from state j to itself A recurrent state j is said to be positive recurrent if jj < and is said to be null-recurrent if jj = In Section 35 it will be seen that null-recurrency can only occur in Markov chains with an in nite state space To illustrate this, consider the Markov chain {Xn } describing the drunkard s walk on an in nite square in Example 311 (N = ) It can be shown for this in nitestate random walk that each state (x, y) is recurrent, but the mean recurrence time of each state is so that all states are null-recurrent The same holds for the in nite-state Markov chain describing the symmetric random walk on the integers (pi,i+1 = pi,i 1 = 1 for any integer i) However, for the symmetric random 2 walk on an in nite lattice in three or more dimensions, the corresponding Markov chain has the property that all states are transient (in three dimensions, the probability of ever returning to the origin when starting there equals 03405) These remarkable results will not be proved here, but are mentioned to show that Markov chains with an in nite state space are intrinsically more complex than nite-state Markov chains

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This section deals with the long-run behaviour of the Markov chain {Xn } In particular, we discuss the characterization of the equilibrium distribution of the process and a formula for the long-run average cost per time unit when a cost structure is imposed on the Markov chain In this section the emphasis is on giving insights into the long-run behaviour of the Markov chain Most of the proofs are deferred to Section 35 331 Preliminaries A natural question for a Markov chain {Xn } is whether the n-step probabilities (n) pij always have a limit as n The answer to this question is negative as shown by the following counterexample Consider a Markov chain with state space I = {1, 2} and one-step transition probabilities pij with p12 = p21 = 1 and (n) p11 = p22 = 0 In this example the n-step transition probabilities pij alternate between 0 and 1 for n = 1, 2, and hence have no limit as n The reason is the periodicity in this Markov chain example In our treatment of Markov chains we will not give a detailed discussion on the relation between the limiting behaviour (n) of the pij and the issue of periodicity The reason is that our treatment of Markov chains emphasizes the study of long-run averages As explained in Section 22, the long-run average behaviour of a stochastic process is in general much easier to handle than its limiting behaviour More importantly, long-run averages are usually required in the analysis of practical applications In the next theorem we prove that for each Markov chain {Xn } the Cesaro limit of the n-step transition probabilities always exists Theorem 331 For all i, j I , limn (1/n) j I, 1 n n lim

if state j is recurrent, if state j is transient,

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(331)

(332)

for any i, j I , where fij is the probability that the process ever makes a transition into state j when the process starts in state i

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