Nonconventional limit theorems in averaging

Yuri Kifer

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 1, page 236-255
  • ISSN: 0246-0203

Abstract

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We consider “nonconventional” averaging setup in the form d X ε ( t ) d t = ε B ( X ε ( t ) , 𝛯 ( q 1 ( t ) ) , 𝛯 ( q 2 ( t ) ) , ... , 𝛯 ( q ℓ ( t ) ) ) where 𝛯 ( t ) , t ≥ 0 is either a stochastic process or a dynamical system with sufficiently fast mixing while q j ( t ) = α j t , α 1 l t ; α 2 l t ; ⋯ l t ; α k and q j , j = k + 1 , ... , ℓ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

How to cite

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Kifer, Yuri. "Nonconventional limit theorems in averaging." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 236-255. <http://eudml.org/doc/272087>.

@article{Kifer2014,
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 $\varXi (t)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_\{j\}(t)=\{\alpha \}_\{j\}t$, $\{\alpha \}_\{1\}&lt;\{\alpha \}_\{2\}&lt;\cdots &lt;\{\alpha \}_\{k\}$ 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.},
author = {Kifer, Yuri},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {averaging; limit theorems; martingales; hyperbolic dynamical systems},
language = {eng},
number = {1},
pages = {236-255},
publisher = {Gauthier-Villars},
title = {Nonconventional limit theorems in averaging},
url = {http://eudml.org/doc/272087},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Kifer, Yuri
TI - Nonconventional limit theorems in averaging
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 1
SP - 236
EP - 255
AB - 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 $\varXi (t)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_{j}(t)={\alpha }_{j}t$, ${\alpha }_{1}&lt;{\alpha }_{2}&lt;\cdots &lt;{\alpha }_{k}$ 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.
LA - eng
KW - averaging; limit theorems; martingales; hyperbolic dynamical systems
UR - http://eudml.org/doc/272087
ER -

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