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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Skip-free Markov chains


Authors: Michael C. H. Choi and Pierre Patie
Journal: Trans. Amer. Math. Soc. 371 (2019), 7301-7342
MSC (2010): Primary 60J10, 60J45, 60J50
DOI: https://doi.org/10.1090/tran/7773
Published electronically: January 24, 2019
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Abstract: The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of the family of fundamental excessive functions, which are defined by means of the theory of the Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to identify the class of skip-free Markov chains whose transition operator has only real and simple eigenvalues. We manage to find a set of sufficient and easy-to-check conditions on the one-step transition probability for a Markov chain to belong to this class. We also study several properties of this class including their spectral expansions given in terms of a Riesz basis, derive a necessary and sufficient condition for this class to exhibit a separation cutoff, and give a tighter bound on its convergence rate to stationarity than existing results.


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Additional Information

Michael C. H. Choi
Affiliation: Institute for Data and Decision Analytics, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, People’s Republic of China
Email: michaelchoi@cuhk.edu.cn

Pierre Patie
Affiliation: School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853
Email: pp396@cornell.edu

DOI: https://doi.org/10.1090/tran/7773
Keywords: Markov chains, potential theory, Martin boundary, fluctuation theory, spectral theory, non-self-adjoint operator, rate of convergence, cutoff
Received by editor(s): January 17, 2017
Received by editor(s) in revised form: March 8, 2018
Published electronically: January 24, 2019
Additional Notes: The second author is grateful for the hospitality of the LMA at the UPPA, where part of this work was completed.
This work was partially supported by NSF Grant DMS-1406599 and by ARC IAPAS, a fund of the Communautée francaise de Belgique.
Article copyright: © Copyright 2019 American Mathematical Society