How to check if a given Markov chain is positive recurrent.

Question

I'm trying to solve a problem which is related to my research, and I have to check whether this infinite-state Markov chain is positive recurrent or not.

Suppose the Markov chain I have has state 0, state 1, ... state n, ... At any given state i, the expected next state is less than i. I know this fact:

E[ S_{i+1} | S_{i} ] - S_{i} < 0.

So, I think this somehow will give me a way to prove that the Markov chain will end up being around state 0, state 1, ... state m, where m is a small value. But, I can't find any method of doing it.

Can anybody have any idea as to how to prove that the Markov chain is positive recurrent?

Thanks.

Answer 1

The condition that $E(S_{i+1}\mid S_i=n)\lt0$ for every $n\ne0$ is not sufficient to guarantee positive recurrence. Counterexamples are discrete Bessel processes of suitable indexes.