 Solution

PutThe length process $L_n=\ell (X_n)$ is a nearest-neighbour chain on the nonnegative integers: away from zero it moves up with probability $q$ and down with probability $p$, while at zero it moves up with probability $q$ and stays put with probability $p$.

Returns of the original chain to the root are exactly returns of $L_n$ to zero, so their recurrence classifications agree. The reflected biased random walk on the nonnegative integers is transient when $q>p$, null recurrent when $q=p$, and positive recurrent when $q<p$. Since $p=1-q$, the conditions are respectivelyIrreducibility transfers the classification from the root to every state.