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

Abstract

Let (Sn) n≥0 be a R d -valued random walk (d ≥ 2). Using Babillot's method [2], 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.

Local limit theorem for a Markov additive process on with a null recurrent internal Markov chain

“…The distribution of an increment is then driven by a Markov chain termed the internal Markov chain. Most results in the context of standard random walks are generalized to Markov additive processes when the Markov operator of the internal chain is assumed to be quasi-compact on a suitable Banach space: among them, a renewal theorem [2,31,32,46], local limit theorem [27,35,37,38,40,47], central limit theorem [27,40], results on the recurrence set [1,41,59], large deviations [51,52], asymptotic expansion of the Green function [45,62], one-dimensional Berry-Essen theorem [27,40,39] with applications to M-estimation, and first passage time [28].…”

Renewal theory for transient Markov chains with asymptotically zero drift

“…Further extensions may be found in [34,36,26]. Renewal theorems for multidimensional random walks may be found in [11], [29], [18] and recent paper [2], see also references therein.…”