= Excursion weights of a reflected birth-death chain
{title2=$w_n=\prod_{r=1}^n p_{r-1}/q_r$}
For a nearest-neighbour chain on the nonnegative integers, reflect at zero with holding probability $q_0$ and upward probability $p_0$. In a recurrent excursion from zero until its first return, expected visits $w_i$ satisfy $w_0=1$ and $w_ip_i=w_{i+1}q_{i+1}$, because each upward crossing is paired with a downward crossing. Thus the expected excursion length is $\sum_iw_i$ by <Tonelli's theorem>. Positive recurrence is equivalent to finiteness of that sum. The hitting-probability scale weights are $\rho_n=\prod_{r=1}^nq_r/p_r$, and $w_n\rho_n=p_0/p_n$ helps show that summable excursion weights imply recurrence.
Back to article page
