This question arose from a queuing theory question. Part of the problem is to prove when the queue is positive recurrent. The jump chain of the queue is a reflecting random walk with the following transition probabilities:
$p_{i,i+2} = \frac{\lambda}{\lambda + \mu}$, $p_{i, i-1} = \frac{\mu}{\lambda + \mu}$ and $p_{0,2} = 1$
I need to prove that the walk is positive recurrent if and only if $2\lambda < \mu$. I know that this can be done by looking at the expected step sizes which is given by $\frac{2\lambda - \mu}{\lambda + \mu}$. This is negative when $2\lambda < \mu$ and so is positive recurrent in that case.
However, other proofs I have seen that try to show recurrence or transience rely on proving that $\sum_{n=1}^{\infty} p^{n}(i,i)$ is infinity or less than infinity. Or even using the definition of positive recurrence which is that $\mathbb{E}_xT_x < \infty$ where $T_x$ is the hitting time to return back to state $x$ after starting at state $x$. How does looking at the expected step size satisfy these definitions of recurrence and transience?
Apologies if this question is trivial or if it has been repeated. Please let me know if this question is missing more details or if I have stated anything incorrectly.
Thank you,