We consider discrete-time birth-death processes with an absorbing state and study the conditional state distribution at time n given that absorption has not occurred by that time but will occur eventually. In particular, we establish conditions for the convergence of these distributions to a proper distribution as n→∞. The problem turns out to be closely related to that of finding conditions for the existence of limits of ratios of n-step transition probabilities as n→∞. Orthogonal polynomials feature in the spectral representation for the n-step transition probabilities of a birth-death process and, consequently, play a key role in the analysis.
van Doorn, E. A., & Schrijner, P. (1995). Ratio limits and limiting conditional distributions for discrete-time birth-death processes. Journal of mathematical analysis and applications, 190(1), 263-284. https://doi.org/10.1006/jmaa.1995.1076