A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension $d\geq3$ and Centered Case

TL;DR

This paper extends the renewal theorem to strongly ergodic Markov chains in dimensions $d extgreater=3$, using spectral methods and weakened moment conditions, broadening its applicability to various ergodic Markov models.

Contribution

It generalizes the renewal theorem for Markov chains in higher dimensions under weaker assumptions, employing spectral techniques and perturbation theory.

Findings

Renewal theorem holds for $d extgreater=3$ Markov chains under new assumptions.

Weak perturbation theorem allows for relaxed moment conditions.

Results apply to $v$-geometrically ergodic, $ ho$-mixing, and Lipschitz Markov models.

Abstract

In dimension $d\geq3$, we present a general assumption under which the renewal theorem established by Spitzer for i.i.d. sequences of centered nonlattice r.v. holds true. Next we appeal to an operator-type procedure to investigate the Markov case. Such a spectral approach has been already developed by Babillot, but the weak perturbation theorem of Keller and Liverani enables us to greatly weaken thehypotheses in terms of moment conditions. Our applications concern the v$-geometrically ergodic Markov chains, the $\rho$-mixing Markov chains, and the iterative Lipschitz models, for which the renewal theorem of the i.i.d. case extends under the (almost) expected moment condition.