Nonconventional limit theorems in averaging
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 236-255.

Nous considérons un cadre non conventionnel de moyenne de la forme $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B\left({X}^{\epsilon }\left(t\right)$, $𝛯\left({q}_{1}\left(t\right)\right),𝛯\left({q}_{2}\left(t\right)\right),...,𝛯\left({q}_{\ell }\left(t\right)\right)\right)$$𝛯\left(t\right)$, $t\ge 0$ est un processus stochastique ou un système dynamique suffisamment mélangeant tandis que ${q}_{j}\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots <{\alpha }_{k}$ et ${q}_{j}$, $j=k+1,...,\ell$ ont une croissance sur-linéaire. Nous montrons que le terme d’erreur après renormalisation est asymptotiquement gaussien.

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B\left({X}^{\epsilon }\left(t\right)$, $𝛯\left({q}_{1}\left(t\right)\right),𝛯\left({q}_{2}\left(t\right)\right),...,𝛯\left({q}_{\ell }\left(t\right)\right)\right)$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while ${q}_{j}\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots <{\alpha }_{k}$ and ${q}_{j}$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

DOI : https://doi.org/10.1214/12-AIHP514
Classification : 34C29,  60F17,  37D20
Mots clés : averaging, limit theorems, martingales, hyperbolic dynamical systems
@article{AIHPB_2014__50_1_236_0,
author = {Kifer, Yuri},
title = {Nonconventional limit theorems in averaging},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
pages = {236--255},
publisher = {Gauthier-Villars},
volume = {50},
number = {1},
year = {2014},
doi = {10.1214/12-AIHP514},
mrnumber = {3161530},
language = {en},
url = {http://archive.numdam.org/item/AIHPB_2014__50_1_236_0/}
}
Kifer, Yuri. Nonconventional limit theorems in averaging. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 236-255. doi : 10.1214/12-AIHP514. http://archive.numdam.org/item/AIHPB_2014__50_1_236_0/

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