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Local limit theorem for a Markov additive process on $\mathbb{Z}^d$ with a null recurrent internal Markov chain

Part of: Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  22 December 2022

Basile de Loynes Open the ORCID record for Basile de Loynes [Opens in a new window]
Basile de Loynes*
Affiliation:
CREST-ENSAI
*
*Postal address: Campus de Ker-Lann, Rue Blaise Pascal, BP 37203, 35172 Bruz Cedex. Email: bdeloynes@math.cnrs.fr
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Abstract

In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. This paper is devoted to a class of inhomogeneous random walks on $\mathbb{Z}^d$ termed ‘Markov additive processes’ (also known as Markov random walks, random walks with internal degrees of freedom, or semi-Markov processes). In this model, the increments of the walk are still independent but their distributions are dictated by a Markov chain, termed the internal Markov chain. While this model is largely studied in the literature, most of the results involve internal Markov chains whose operator is quasi-compact. This paper extends two results for more general internal operators: a local limit theorem and a sufficient criterion for their transience. These results are thereafter applied to a new family of models of drifted random walks on the lattice $\mathbb{Z}^d$ .

Keywords

Inhomogeneous random walkssemi-Markov processesMarkov random walksrandom walks with internal degrees of freedomrecurrencetransienceFourier analysis

MSC classification

Primary: 60F15: Strong theorems 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60J45: Probabilistic potential theory 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J55: Local time and additive functionals
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 570 - 588
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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