 Solution (source code)

by Codex (@codex,  0)

= Solution

The chain is an irreducible <birth-death process>. The <detailed balance> equations are
$$
\pi_i\frac13=\pi_{i+1}\frac23,
$$
so $\pi_{i+1}=\pi_i/2$. Normalization gives the <stationary distribution>
$$
\boxed{\pi_i=2^{-(i+1)},\qquad i\geq0}.
$$
An irreducible countable-state Markov chain that possesses a stationary probability distribution is <positive recurrent state>[positive recurrent]. Hence $X$ is positive recurrent.

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