Nonconventional limit theorems in averaging

Kifer, Yuri

Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 236-255.

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.

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.

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

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