A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension d ≥ 3 and Centered Case

Abstract

In dimension d ≥ 3, we present a general assumption under which the renewal theorem established by Spitzer (1964) 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 (Ann Inst Henri Poincaré, Sect B, Tome 24(4):507-569, 1988), but the weak perturbation theorem of Keller and Liverani (Ann Sc Norm Super Pisa CI Sci XXVIII(4):141-152, 1999) enables us to greatly weaken the moment conditions of Babillot (Ann Inst Henri Poincaré, Sect B, Tome 24(4):507-569, 1988). Our applications concern the vgeometrically ergodic Markov chains, the ρ-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.

Limit theorems for stationary Markov processes with L2-spectral gap

“…Other limit theorems can be stated under Condition (AS2) as, for instance, a multidimensional Berry-Esseen theorem in the Prohorov metric (see [49,Sect. 9]), and the multidimensional renewal theorems (see [39]). Although Proposition 4.2 extends to the case when the order of regularity m 0 is not integer, it does not allow to deal with the convergence of Y n (properly normalized) to stable laws, since we assume α > m 0 (in place of the expected condition α = m 0 ).…”

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

“…Under Hypothesis H Lim , we have ξ(·) ≤ ξ(x 0 ) + S(1 + d(·, x 0 )) s+1 , so that ξ(·) is π-integrable under Conditions (40) (41), see [7,3]. The others conditions of Hypothesis (H) follow from the results proved in [13] (centered case). Indeed, note that ξ c = ξ − m is π-centered.…”

“…Indeed, note that ξ c = ξ − m is π-centered. Then the existence of Σ c := lim n 1 n E µ [S ⊗2 n,c ] is proved in [13,Sect. 4] under the three above hypotheses.…”

“…To define the renewal measure U a (·), the sequence (S n ) n≥0 has to be transient: for independent or Markov random walks, this leads to consider the following cases: The behavior of U a (·) also depends on the usual lattice or non-lattice conditions. This work is the continuation of [11] (case d = 1) and [13] (centered case in dimension d ≥ 3). More specifically, in this paper, we consider the non-centered case in dimension d ≥ 2, and we present some general assumptions involving the characteristic function of S n , under which we have the same conclusion as in the classical renewal theorem for random walks with independent and identically distributed (i.i.d.)…”

Théorème de renouvellement pour chaînes de Markov fortement ergodiques : application aux modèles itératifs lipschitziens

“…En particulier, notre méthode devrait permettre d'étendre les possibilités d'applications en géométrie ergodique, comme par exemple dans [4] pour l'étude du comportement asymptotique de fonctions orbitales de certains groupes discrets. Par ailleurs, en utilisant [14], les théorèmes de renouvellement multi-dimensionnels de [2] peuvent être également améliorés [6].…”