Biased random walk

A biased nearest-neighbor walk steps right and left with unequal probabilities.

Let a discrete-time chain move up with probability $q$ and down with probability $p=1-q$ away from zero, while at zero it moves up with probability $q$ and stays with probability $p$. It is transient for $q>p$, null recurrent for $q=p$, and positive recurrent for $q<p$. In the positive-recurrent case its invariant distribution isand aperiodicity from the self-loop at zero implies convergence to this distribution.