Nonconventional limit theorems in averaging

Kifer, Yuri

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 236-255 / Harvested from Numdam

Résumé

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 $\varXi (t)$, $t\ge 0$ 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.

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 $q_{j}(t)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots <{\alpha }_{k}$ 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.

Détail

Publié le : 2014-01-01

MR 3161530

DOI : https://doi.org/10.1214/12-AIHP514
Classification:  34C29,  60F17,  37D20

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