It is well known that an one dimensional random walk with probability $p$ to the right and probability $q$ to the left is transient if $p\neq q$ and recurrent if $p=q$. However, for the more general case when $p$ and $q$ may depend on the location $i$, I am wondering whether there's any clean way to tell if the states are recurrent or not. In particular, I am interested in the scenario when $$\frac{p_{i, i+1}}{p_{i, i-1}} \to 1$$ as $i$ tends towards positive infinity.
My current guess is that under this scenario whether the states are recurrent or not depends on the rate of the convergence. Yet I struggled to come up with any rigorous statement. I would appreciate if someone could provide some ideas.