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

Let $(S_n{)}_n$ be a $R^d$-valued random walk ($d\ge 2$). 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.