One closed class

Topic: Classes, recurrence, and reversibility · more questions on this topic
A walk on a path of four vertices ›

A chain on \{1,2,3\} moves from 1 to 2 with probability 1, from 2 to 1 with probability 1/2 and from 2 to 3 with probability 1/2, and from 3 to 3 with probability 1. Find the communicating classes, the closed classes, and the stationary distributions.

Worked answer (click to show)

The transition matrix is

P=\begin{pmatrix}0&1&0\\ 1/2&0&1/2\\ 0&0&1\end{pmatrix}.

States 1 and 2 communicate: each is one step from the other. State 3 is accessible from 2, and nothing leaves 3, so 3 is not accessible back to \{1,2\}. The communicating classes are \{1,2\} and \{3\}. Only \{3\} is closed. States 1 and 2 are transient. The only stationary distribution is (0,0,1), the point mass on the single closed class.