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Preprints, Working Papers, ... Year : 2020

Local Limit Theorem for Additive Processes on $\mathbb Z^d$ with Null Recurrent Internal Markov Chain

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 Degree 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. Whereas 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 $\mathbb Z^d$.
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Dates and versions

hal-01274873 , version 1 (16-02-2016)
hal-01274873 , version 2 (12-05-2016)
hal-01274873 , version 3 (13-06-2016)
hal-01274873 , version 4 (18-01-2018)
hal-01274873 , version 5 (07-08-2020)
hal-01274873 , version 6 (22-06-2022)

Licence

Attribution - NonCommercial - NoDerivatives

Identifiers

  • HAL Id : hal-01274873 , version 6

Cite

Basile de Loynes. Local Limit Theorem for Additive Processes on $\mathbb Z^d$ with Null Recurrent Internal Markov Chain. 2020. ⟨hal-01274873v6⟩
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