### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 047, 9 pages      arXiv:1801.00506      https://doi.org/10.3842/SIGMA.2018.047
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### On the Strong Ratio Limit Property for Discrete-Time Birth-Death Processes

Erik A. van Doorn
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Received January 03, 2018, in final form May 13, 2018; Published online May 15, 2018

Abstract
A sufficient condition is obtained for a discrete-time birth-death process to possess the strong ratio limit property, directly in terms of the one-step transition probabilities of the process. The condition encompasses all previously known sufficient conditions.

Key words: (a)periodicity; birth-death process; orthogonal polynomials; random walk measure; ratio limit; transition probability.

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