Stationary weights of a reflected birth-death chain ($w_0=1,w_k={\prod }_{j=0}^{k-1}{u}_j/d_{j+1}$)

Consider an irreducible Markov chain on the nonnegative integers with up-step probabilities $u_k>0$ and down-step probabilities $d_k>0$ for $k\ge 1$, 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 ${\pi }_0{u}_0-{\pi }_1{d}_1$ to zero, and the remaining stationary equations make the flow constant along all edges. Hence positive recurrence is equivalent to ${\sum }_k{w}_k<\infty$, and then ${\pi }_k=w_k/{\sum }_j{w}_j$. This also allows holding probabilities of a Markov chain.