Ratio Limits and Limiting Conditional Distributions for Discrete-Time Birth-Death Processes

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.