Nonconventional limit theorems in averaging

TL;DR

This paper establishes that in a nonconventional averaging setup involving fast mixing processes and multiple time scales, the normalized error term converges to a Gaussian distribution, extending classical averaging results.

Contribution

It proves the asymptotic Gaussianity of the error term in nonconventional averaging with multiple time scales and fast mixing processes.

Findings

Normalized error term is asymptotically Gaussian.

Results extend classical averaging to nonconventional setups.

Applicable to processes with multiple growth rates.

Abstract

We consider "nonconventional" averaging setup in the form $\frac {dX^\epsilon(t)}{dt}=\epsilon B\big(X^\epsilon(t),\xi(q_1(t)), \xi(q_2(t)),...,\xi(q_\ell(t))\big)$ where $\xi(t),t\geq 0$ is either a stochastic process or a dynamical system (i.e. then $\xi(t)=F^tx$) with sufficiently fast mixing while $q_j(t)=\al_jt,\,\al_1<\al_2<...<\al_k$ 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.