3
$\begingroup$

Suppose $X_n$ is an irreducible, aperiodic and Harris recurrent Markov chain. It is well known that in this case, $X_n$ has a stationary distribution $\pi$.

Are there any conditions that are sufficient to guarantee that $\pi$ is finite? In other words, what criteria are there to show that the chain is positive recurrent and not null-recurrent?

$\endgroup$
4
$\begingroup$

Let's call your state space $\mathcal{X}$. Then, you want to be able to find a function $V \colon \mathcal{X} \to \mathbf{R}_+$ and a small set $C$ such that $$ PV \le V - 1 + K \mathbf{1}_C\;, $$ where $P$ is your Markov operator, $K$ is some constant, and $\mathbf{1}_C$ denotes the indicator function of $C$. Since $V$ bounds the expectation of the return time to $C$, this is what you want. This condition is actually both sufficient and necessary since the expected return time to $C$ does satisfy that bound. In some cases, such a Lyapunov function $V$ can easily be guessed, in other cases it is more of an artform to engineer a good one. (See https://arxiv.org/abs/0810.5431 for a class of very interesting examples in continuous time.)

$\endgroup$

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.