Geometric-jump countdown Markov chain
= Geometric-jump countdown Markov chain
{title2=$p_{i,i-1}=1,\quad p_{0j}=pq^{j-1}$}
On the nonnegative integers, every positive state decreases by one, while zero jumps to $j\geq1$ with probability $pq^{j-1}$, $p=1-q\in(0,1)$. Its return time from zero is one plus this geometric jump, so it has mean $(1+p)/p$. Return lengths two and three occur, establishing <aperiodicity>; deterministic downward motion and positive jumps establish irreducibility.