Multidimensional renewal theory in the non-centered case. Application to strongly ergodic Markov chains

TL;DR

This paper extends renewal theory to multidimensional, non-centered random walks and applies it to additive functionals of strongly ergodic Markov chains, under specific conditions on their characteristic functions.

Contribution

It generalizes renewal theorems to non-i.i.d. multidimensional cases and applies these results to ergodic Markov chains with minimal moment assumptions.

Findings

Renewal theorem holds for multidimensional non-centered random walks.

Conditions on characteristic functions ensure the renewal property.

Application to strongly ergodic Markov chains under non-lattice and moment conditions.

Abstract

Let $(S_n)_n$ be a $R^d$-valued random walk ($d\geq2$). Using Babillot's method [2], we give general conditions on the characteristic function of $S_n$ under which $(S_n)_n$ satisfies the same renewal theorem as the classical one obtained for random walks with i.i.d. non-centered increments. This statement is applied to additive functionals of strongly ergodic Markov chains under the non-lattice condition and (almost) optimal moment conditions.