 Reflected chain with geometric downward jumps

Fix $0<p<1$ and put $q=1-p$. For independent geometric random variables $D_n$ with $\mathbb{P}(D_n=k)=p{q}^k$ for $k\ge 0$, the Reflected chain with geometric downward jumps obeys $X_{n+1}=\max (0,X_n+1-D_n)$. Its transition probabilities are $P_{ij}=p{q}^{i-j+1}$ for $1\le j\le i+1$ and $P_{i0}=q^{i+1}$.

The 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. Before reflection the mean increment is $1-\mathbb{E}{D}_n=(2p-1)/p$: downward drift corresponds to $p<1/2$.

The chain is positive recurrent for $p<1/2$, with Stationary distribution ${\pi }_j=(1-p/q)(p/q{)}^j$ for $j\ge 0$. The ratio $p/q<1$ ensures these weights sum to one. It is null recurrent at $p=1/2$ and transient for $p>1/2$. The parameter restriction matters: $p=0$ does not define the displayed geometric distribution, and $p=1$ produces deterministic upward motion rather than an irreducible chain.