A chain that never settles

Topic: Markov chains · more questions on this topic
‹ Any two-state chain A walk with a floor ›

A chain on \{1,2\} moves to the other state at every step. Find P^n for every n, and a probability vector \boldsymbol\pi with \boldsymbol\pi=\boldsymbol\pi P. Explain why P^n has no limit.

Worked answer (click to show)

P=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad P^2=I,

so P^n=I for even n and P^n=P for odd n. The vector \boldsymbol\pi=(1/2,1/2) satisfies \boldsymbol\pi P=\boldsymbol\pi. But (P^n)_{11} alternates between 1 and 0, so it has no limit. The chain is irreducible but has period 2: returns to a state happen only at even times. A stationary distribution exists; a limiting one does not.