Nonconventional limit theorems in averaging

Kifer, Yuri

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 $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ où $𝛯\left(t\right)$, $t\ge 0$ 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 $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ 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.

Classification : 34C29, 60F17, 37D20
Mots clés : averaging, limit theorems, martingales, hyperbolic dynamical systems

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