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Local Limit Theorem for Additive Processes with Null Recurrent Internal Markov Chain

Résumé : 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 involves internal Markov chains whose operators are quasi-compact. This paper extends two results for more general internal operator: 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 $\mathbb Z^d$. Incidentally, in the examples under consideration, we give a new bound on the rate of convergence in the Wasserstein metric of appropriate scaling of Random Walks in Random Scenery.
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Submitted on : Friday, August 7, 2020 - 5:40:12 PM
Last modification on : Wednesday, August 12, 2020 - 3:34:52 AM

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Basile de Loynes. Local Limit Theorem for Additive Processes with Null Recurrent Internal Markov Chain. 2020. ⟨hal-01274873v5⟩

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