From Chpater 2 of Introduction to Stochastic Processes Hoel

From Chpater 2 of Introduction to Stochastic Processes -Hoel

3 Let Pi be a stationary distribution of a Markov chain. Show that if Pi(x) > 0 and x leads to y, then Pi(y) > 0.

Solution

As pi is a stationary distribution of Markov chain, the following defintion holds good.

Definition: A (discrete-time) stochastic process {Xn : n ? 0} is stationary if for any time points i1, . . . , in and any m ? 0, the joint distribution of (Xi1 , . . . , Xin ) is the same as the joint distribution of (Xi1+m, . . . , Xin+m). So