Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line

In this paper, we study (1,2) and (2,1) random walks in spatially inhomogeneous environments on the lattice of positive half line. We assume that the transition probabilities at site n are asymptotically constant as n → ∞ . We get some elaborate limit behaviors of various escape probabilities and hi...

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Bibliographic Details
Published in:Journal of theoretical probability 2024-03, Vol.37 (1), p.409-445
Main Authors: Tang, Lanlan, Wang, Hua-Ming
Format: Article
Language:English
Subjects:
Asymptotic properties
Markov processes
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Random walk
Statistics
Transition probabilities
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Summary:In this paper, we study (1,2) and (2,1) random walks in spatially inhomogeneous environments on the lattice of positive half line. We assume that the transition probabilities at site n are asymptotically constant as n → ∞ . We get some elaborate limit behaviors of various escape probabilities and hitting probabilities of the walk. Such observations and some delicate analysis of continued fractions and the product of nonnegative matrices enable us to give criteria for finiteness of the number of cutpoints of both (1,2) and (2,1) random walks, which generalize Csáki et al. (J Theor Probab 23:624–638, 2010) and Wang (Markov Process Relat Fields 25:125–148, 2019). For near-recurrent random walks, we also study the asymptotics of the number of cutpoints in [0,  n ].
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-023-01293-2