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February 2014 Nonconventional limit theorems in averaging
Yuri Kifer
Ann. Inst. H. Poincaré Probab. Statist. 50(1): 236-255 (February 2014). DOI: 10.1214/12-AIHP514

Abstract

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}X^{\varepsilon }(t)}{\mathrm{d}t}=\varepsilon B(X^{\varepsilon }(t)$, $\varXi (q_{1}(t)),\varXi (q_{2}(t)),\ldots,\varXi (q_{\ell}(t)))$ where $\varXi (t)$, $t\geq0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_{j}(t)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots<{\alpha }_{k}$ and $q_{j}$, $j=k+1,\ldots,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

Nous considérons un cadre non conventionnel de moyenne de la forme $\frac{\mathrm{d}X^{\varepsilon }(t)}{\mathrm{d}t}=\varepsilon B(X^{\varepsilon }(t)$, $\varXi (q_{1}(t)),\varXi (q_{2}(t)),\ldots,\varXi (q_{\ell}(t)))$ où $\varXi (t)$, $t\geq0$ est un processus stochastique ou un système dynamique suffisamment mélangeant tandis que $q_{j}(t)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots<{\alpha }_{k}$ et $q_{j}$, $j=k+1,\ldots,\ell$ ont une croissance sur-linéaire. Nous montrons que le terme d’erreur après renormalisation est asymptotiquement gaussien.

Citation

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Yuri Kifer. "Nonconventional limit theorems in averaging." Ann. Inst. H. Poincaré Probab. Statist. 50 (1) 236 - 255, February 2014. https://doi.org/10.1214/12-AIHP514

Information

Published: February 2014
First available in Project Euclid: 1 January 2014

zbMATH: 1353.37107
MathSciNet: MR3161530
Digital Object Identifier: 10.1214/12-AIHP514

Subjects:
Primary: 34C29
Secondary: 37D20 , 60F17

Keywords: averaging , Hyperbolic dynamical systems , limit theorems , Martingales

Rights: Copyright © 2014 Institut Henri Poincaré

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Vol.50 • No. 1 • February 2014
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