Parity balance on entry to a period-two Markov class

Suppose a Markov chain leaves transient states and enters a closed two-state class which alternates deterministically. If entry is always into the first state, let $a$ and $b$ be the probabilities of entering at even and odd times. After the probability of still being transient tends to zero, the even-time probabilities at the two recurrent states tend to $a,b$, while the odd-time probabilities tend to $b,a$. Individual limits therefore exist exactly when $a=b$. A periodic recurrent class can thus have convergent state probabilities from particular transient starting states even though convergence fails from its own states.

// Target: geometry-and-topology.bigb