= Stationary weights of a reflected birth-death chain
{title2=$w_0=1,\quad w_k=\prod_{j=0}^{k-1}u_j/d_{j+1}$}
Consider an <irreducible Markov chain> on the nonnegative integers with up-step probabilities $u_k>0$ and down-step probabilities $d_k>0$ for $k\geq1$, reflected at zero. The displayed weights satisfy <detailed balance>. Every <stationary distribution> is proportional to them: the stationary equation at zero sets the edge flow $\pi_0u_0-\pi_1d_1$ to zero, and the remaining stationary equations make the flow constant along all edges. Hence <positive recurrence> is equivalent to $\sum_kw_k<\infty$, and then $\pi_k=w_k/\sum_jw_j$. This also allows <holding probabilities of a Markov chain>.
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