Harmonic functions and stationary distributions for asymptotically homogeneous transition kernels on Z⁺

2013/12/08 by Denis Denisov, Denisov, Denis, Dmitry Korshunov +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #31C05 #39A10 #60F10 #60J10 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:31C05 #msc:39A10 #msc:60F10 #msc:60J10

paper · pdf · doi:10.48550/arxiv.1312.2201

arxiv created 2013/12/08 · openalex publication_date 2013/12/08 · arxiv updated 2013/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We suggest a method for constructing positive harmonic functions for a wide class of transition kernels on Z⁺. We also find natural conditions under which these functions have positive finite limits at infinity. Further, we apply our results on harmonic functions to asymptotically homogeneous Markov chains on Z⁺ with asymptotically negative drift. More precisely, assuming that Markov chain satisfy Cramér's condition, we study the tail asymptotics of the stationary distribution. In particular, we clarify the influence of the rate of convergence of jumps of the chain towards the limiting distribution.

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