# Does a solution for the stationary distribution of a Markov chain guarantee the distribution exists?

Say we have a Markov chain with a countably infinite state space, e.g. the non-negative integers.

If we can form and solve equations for the stationary distribution {$\pi_i$}, that satisfies:

$\pi_i = \sum_{j} P_{ji}\pi_j , \ \ \ \ \ i \in \{0,1,2,...\}$

$\sum_{j} \pi_j = 1$

Is it guaranteed that the stationary distribution we've solved for exists?

Or could we solve these equations even if the chain were null recurrent?

• Much appreciated, and sorry to be dense ni6go... would I be right thinking then that if the states were not positive recurrent, I would have been unable to find {$\pi$} that solved both conditions (above)? – conjectures May 3 '13 at 19:09