Let $X$ be a random walk on $\mathbb{N}^*:=\mathbb{N}\cup\left\{0\right\}$ with the following transition probability
$\displaystyle p(n,n+k)=\frac{p^k}{Z_n}$ for $n,k\in \mathbb{N}^*$,
$\displaystyle p(n,n-k)=\frac{1}{Z_n\cdot k}$ for $1\le k\in n\in \mathbb{N}$,
where $p\in (0,1)$ is a fixed constant and $Z_n$ is the normalizing constant that ensures the $p(n,\cdot)$ is a probability kernel for each $n\in\mathbb{N}^*.$
How to prove that $X$ is a positive recurrent chain? The hints says that positive recurrent criterion for Harris chains can also be applied here. But I do not know how to apply the criterion to this random walk?
The transition probability is not nice for me to apply the recurrent criterion for a random walk (e.g. Chung-Fuchs, etc.) because the form is not that nice. I am writing to ask are there some neat ways to deal with this kind of random walk?