Positive multiplications in difference equations: With an appendix by David R. McDonald
The authors study the generalizations of three-term recurrence relations with respect to positivity and what amounts to a boundary for the corresponding Markov chains. They show that an eventual ``linearization'' property holds. They also obtain limit ratio theorems for many of the random walks/Markov chains. The techniques used here are a mixture of algebra and functional analysis, with some ideas from the version of Choquet theory developed for dimension groups by \textit{E. G. Effros}, \textit{D. E. Handelman}, and \textit{C.-L. Shen} [Am. J. Math. 102, 385-407 (1980; Zbl 0457.46047)] and others. In the end of the paper there is an appendix by D. R. McDonald giving fairly general conditions under which stationary Markov chains on the positive integers have trivial Poisson boundary. This is used in the main part of the paper to treat a larger class of recurrence relations.
scientific article; zbMATH DE number 763975