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

Lanlan Tang and Hua-Ming Wang ()
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Lanlan Tang: Anhui Normal University
Hua-Ming Wang: Anhui Normal University

Journal of Theoretical Probability, 2024, vol. 37, issue 1, 409-445

Abstract: Abstract 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 $$n\rightarrow \infty .$$ 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].

Keywords: Random walk; Cutpoints; Product of nonnegative matrices; Continued fractions; 60J10; 11A55 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01293-2

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