Nonconventional limit theorems in averaging

Abstract

We consider "nonconventional" averaging setup in the form $\frac {dX^ε(t)}{dt}=εB\big(X^ε(t),ξ(q_1(t)), ξ(q_2(t)),...,ξ(q_\ell(t))\big)$ where $ξ(t),t\geq 0$ is either a stochastic process or a dynamical system (i.e. then $ξ(t)=F^tx$) 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.