Does a stationary distribution exist for this MC?

Question

Let a Markov Chain be given with the following transition probabilities; $$p_{0,0} = \frac{1}{2}, p_{0,1} = \frac{1}{2}, P_{i, i+1} = \frac{i + 1}{i+2} (i \geq 1), p_{i,0} = \frac{1}{i+2} (i \geq 1)$$

I'm trying to figure out whether or not a stationary distribution exists for it. I've tried to show $0$ is a transient state and hence all states are transient (irreducibility), thus the MC wouldn't have a sationary distribution. But all the states are recurrent, so I don't know where to go from here!

How can I be sure if a M.C has a stationary distribution or not? I know that for a FINITE state space, irreducible, aperiodic M.C, one exists.

How do I go about this in the countably infinite state space case?

Answer 1

Hint 1:

Recall that for an irreducible chain (such as this one) having a stationary distribution is equivalent to a Markov Chain being null recurrent -- that is, $\mathbb E_x[T_x] < \infty$ for all $x$ in the state space.

Hint 2:

Recall that null recurrence is "infectious" -- that is, $\mathbb E_x[T_x] < \infty \implies \mathbb E_y[T_y] < \infty$ for all other $y$ in the state space.

Hint 3:

You can fairly easily compute $\mathbb E_0 [T_0]$ directly by considering the possible excursions from $0$.