Multidimensional Renewal Theory in the Non-Centered Case. Application to Strongly Ergodic Markov Chains

Abstract

Let (Sn)n ≥ 0 be a ℝd-valued random walk (d ≥ 2). Using Babillot’s method (Babillot, Ann Inst Henri Poincare, B, Tome 24(4):507–569, 1988), we give general conditions on the characteristic function of Sn under which (Sn)n ≥ 0 satisfies the same renewal theorem as in the independent case (i.e. the same conclusion as in the case when the increments of (Sn)n ≥ 0 are assumed to be independent and identically distributed). This statement is applied to additive functionals of strongly ergodic Markov chains under the non-lattice condition and (almost) optimal moment conditions.