Ratio limits and limiting conditional distributions for discrete-time birth-death processes
The authors study discrete-time birth-death processes with an absorbing state by considering the corresponding random walk polynomials for the \(n\)-step transition probabilities. Conditions are given for existence of limits of ratios of \(n\)-step transition probabilities for \(n \to \infty\). The conditional state distribution at time \(n\), given that absorption has not occurred by that time but will occur eventually, is studied. Conditions are given in which these distributions converge to a proper distribution. Finally, random walks related to Jacobi polynomials and random walks with almost constant parameters are studied.
scientific article; zbMATH DE number 765123