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

Loïc Hervé; Françoise Pène

Bulletin de la Société Mathématique de France (2010)

  • Volume: 138, Issue: 3, page 415-489
  • ISSN: 0037-9484

Abstract

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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 Ł 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.

How to cite

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Hervé, Loïc, and Pène, Françoise. "The Nagaev-Guivarc’h method via the Keller-Liverani theorem." Bulletin de la Société Mathématique de France 138.3 (2010): 415-489. <http://eudml.org/doc/272519>.

@article{Hervé2010,
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 $Ł^2$-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 $\varepsilon &gt;0$), to those of the i.i.d. case.},
author = {Hervé, Loïc, Pène, Françoise},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Markov chains; central limit theorems; edgeworth expansion; spectral method},
language = {eng},
number = {3},
pages = {415-489},
publisher = {Société mathématique de France},
title = {The Nagaev-Guivarc’h method via the Keller-Liverani theorem},
url = {http://eudml.org/doc/272519},
volume = {138},
year = {2010},
}

TY - JOUR
AU - Hervé, Loïc
AU - Pène, Françoise
TI - The Nagaev-Guivarc’h method via the Keller-Liverani theorem
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 3
SP - 415
EP - 489
AB - 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 $Ł^2$-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 $\varepsilon &gt;0$), to those of the i.i.d. case.
LA - eng
KW - Markov chains; central limit theorems; edgeworth expansion; spectral method
UR - http://eudml.org/doc/272519
ER -

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