Nonconventional limit theorems in averaging

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 $\xi (t)=F^{t}x$) with sufficiently fast mixing while $q_j(t)={\text{al}}_{j}t,{\text{al}}_1<{\text{al}}_2<\mathrm{.}\mathrm{.}\mathrm{.}<{\text{al}}_k$ and $q_j,j=k+1,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\ell }$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.