Reflected chain with geometric downward jumps ($X_{n+1}=\max (0,X_n+1-D_n)$)

Let independent $D_n$ have $\mathbb{P}(D_n=k)=p{q}^k$, $k\ge 0$, $q=1-p$. The displayed chain has transition probabilities $P_{ij}=p{q}^{i-j+1}$ for $1\le j\le i+1$ and $P_{i0}=q^{i+1}$. Its probability of hitting zero from $i\ge 1$ is one when $p\le 1/2$ and $(q/p{)}^{i+1}$ when $p>1/2$. The first-step boundary equation at $i=1$ is essential for the extra exponent. The chain is positive recurrent for $p<1/2$, with stationary probabilities $(1-p/q)(p/q{)}^j$, null recurrent at $p=1/2$, and transient for $p>1/2$.