Imagine a chain where we start in some state (call it state zero), and the probability of being in state zero after transitions is zero unless is a multiple of some number, say . If is the smallest such number, then we say that that state has period . We’ve already seen an example of a chain with such a state in the simple random walk. In the simple random walk it was impossible to move from any state back to itself in an odd number of transitions: to do so you have to step left the same number of times you step right, which is of course an even number. Since two divides every even number, we have that the state has a period of two. A state with period one is called aperiodic.

If a state is recurrent, we say that the state is positive recurrent if the expected amount of time between recurrences (when the chain is in that state) is finite. A state that is recurrent but not positive recurrent is called null recurrent. A positive recurrent, aperiodic state is called ergodic.

An example of a null recurrent state appears in the simple random walk. We’ve seen that all states are recurrent, but it turns out that all states are in fact null recurrent. For a discussion of the null recurrence of states in the simple random walk, see section 2.2 in The Theory of Stochastic Processes by Cox and Miller.

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