= Reflected chain with geometric downward jumps
{title2=$X_{n+1}=\max(0,X_n+1-D_n)$}
Fix $0<p<1$ and put $q=1-p$. For <independent> <geometric random variable>[geometric random variables] $D_n$ with $\mathbb P(D_n=k)=pq^k$ for $k\geq0$, the <Reflected chain with geometric downward jumps> obeys $X_{n+1}=\max(0,X_n+1-D_n)$. Its <transition probabilities> are $P_{ij}=pq^{i-j+1}$ for $1\leq j\leq i+1$ and $P_{i0}=q^{i+1}$.
The <Probability> of hitting zero from $i\geq1$ is one when $p\leq1/2$ 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 Markov chain>[positive recurrent] for $p<1/2$, with <Stationary distribution> $\pi_j=(1-p/q)(p/q)^j$ for $j\geq0$. The ratio $p/q<1$ ensures these weights sum to one. It is <null recurrent Markov chain>[null recurrent] at $p=1/2$ and <transient Markov chain>[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 Markov chain>[irreducible] chain.
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