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

Guibourg, Denis

doi:10.1016/j.crma.2008.02.010

Cited by 3 publications

(7 citation statements)

References 12 publications

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“…In recent works [43,45,14,38,46,33,31], a new procedure, based on the perturbation the-orem of Keller-Liverani [52] (see also [5] p. 177), allows to get round the previous difficulty and to greatly improve the Nagaev-Guivarc'h method when applied to unbounded functionals ξ. Our work outlines this new approach, and presents the applications, namely : a multidimensional local limit theorem, a one-dimensional Berry Esseen theorem, a first-order Edgeworth expansion.…”

Section: Introduction Setting and Notationsmentioning

confidence: 99%

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The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Hervé¹,

Pène²

2010

Bul. Soc. Math. France

158

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Abstract. -The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order Edgeworth expansion, and a multidimensional Berry-Esseen type theorem in the sense of the Prohorov metric. When applied to the exponentially L 2 -convergent Markov chains, to the v-geometrically ergodic Markov chains and to the iterative Lipschitz models, the three first above cited limit theorems hold under moment conditions similar, or close (up to ε > 0), to those of the i.i.d. case. Résumé

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Section: Introduction Setting and Notationsmentioning

confidence: 99%

“…Extensions. The operator-type derivation procedure (B) may be also used to investigate renewal theorems [33] [34], and to study the rate of convergence of statistical estimators for strongly ergodic Markov chains (thanks to the control of the constants in (B)), see [47].…”

Section: Introduction Setting and Notationsmentioning

confidence: 99%

The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Hervé¹,

Pène²

2010

Bul. Soc. Math. France

158

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“…Recall that w(·) is defined in (11). In the two next propositions, we consider any real numbers m > m d and r > 0.…”

Section: Study Of the Second Error Term E 1 (A)mentioning

confidence: 99%

“…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.)…”

Section: Introductionmentioning

confidence: 99%

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

Guibourg

Hervé

2012

Potential Anal

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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.

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“…This new approach, in which the Keller-Liverani perturbation theorem [18,19,10] plays a central role, is fully described in [17] and applied to prove some usual refinements of the central limit theorem (CLT). It is used in [11] to establish a one-dimensional (non-centered) Markov renewal theorem. Also mention that one of the more beautiful applications of this method concerns the convergence to stable laws, see [13,5,9].…”

Section: Introductionmentioning

confidence: 99%

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

Guibourg

Hervé

2010

Potential Anal

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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.

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

Cited by 3 publications

References 12 publications

The Nagaev-Guivarc’h method via the Keller-Liverani theorem

The Nagaev-Guivarc’h method via the Keller-Liverani theorem

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

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

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