Excursion weights of a reflected birth-death chain ($w_n={\prod }_{r=1}^n{p}_{r-1}/q_r$)

Consider an irreducible Birth-death chain on $\{0,1,2,\dots \}$. At zero let $P(0,1)=p_0>0$ and $P(0,0)=1-p_0$. At $i\ge 1$, let $P(i,i+1)=p_i>0$ and $P(i,i-1)=q_i>0$, with $p_i+q_i=1$. DefineIn the recurrent case, $w_i$ is the expected number of visits to $i$ during a cycle started at zero, counting times $0\le m<{\tau }_0^{+}$. The initial visit gives $w_0=1$. Each upward crossing of an edge before the return is paired with a downward crossing, giving $w_i{p}_i=w_{i+1}{q}_{i+1}$ and the product formula. Tonelli's theorem then gives ${\mathbb{E}}_0{\tau }_0^{+}={\sum }_i{w}_i$. Consequently the recurrent chain is positive recurrent exactly when this sum is finite, and its Stationary distribution is $w_i/{\sum }_j{w}_j$.

The finiteness criterion also rules out transience without assuming recurrence in advance. Put ${\rho }_n={\prod }_{r=1}^n{q}_r/p_r$ and ${\rho }_0=1$. Then $w_n{\rho }_n=p_0/p_n\ge p_0$. If ${\sum }_n{w}_n<\infty$, then $w_n\to 0$, so ${\rho }_n\to \infty$ and ${\sum }_n{\rho }_n=\infty$. The birth-death scale-function hitting formula therefore implies recurrence. This separates stationary occupation weights $w_n$ from the inverse-direction scale weights ${\rho }_n$, which serve different purposes.