How do i determine the number of stationary distributions that a Markov chain has if it is not irreducible or regular.

The transition matrix is

1     0    0
0.25  0.5  0.25
0     0    1

is there infinitely many?


1 Answer 1


Yes, exactly. We have two absorbing states : the first and the third. If process starts in the first state, it stays here forewer. And the same for the third state. So both probability vectors $\pi_1=(1,0,0)$ (row vector) and $\pi_3=(0,0,1)$ provide stationary distributions : $\pi_1P=\pi_1$ and $\pi_3P=\pi_3$. And any linear combination of these vectors $\pi=p\pi_1+(1-p)\pi_3$ with $0\leq p\leq 1$ provides stationary distribution too: $$ \pi P= p\pi_1P+(1-p)\pi_3P =p\pi_1+(1-p)\pi_3 = \pi. $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.