Some Strong Limit Theorems in Averaging

The paper deals with the fast-slow motions setups in the discrete time \(X^{\varepsilon }((n+1){\varepsilon })=X^{\varepsilon }(n{\varepsilon })+{\varepsilon }B(X^{\varepsilon }(n{\varepsilon }),\xi (n))\) $X^{\epsilon }((n+1)\epsilon )=X^{\epsilon }(n\epsilon )+\epsilon B(X^{\epsilon }(n\epsilon ),\xi (n))$ , \(n=0,1,...,[T/{\varepsilon }]\) $n=0,1,...,[T/\epsilon ]$ and the continuous time \(\frac{dX^{\varepsilon }(t)}{dt}=B(X^{\varepsilon }(t),\xi (t/{\varepsilon })),\, t\in [0,T]\) $\frac{d{X}^{\epsilon }(t)}{\mathit{dt}}=B(X^{\epsilon }(t),\xi (t/\epsilon )),t\in [0,T]$ where B is a smooth in the first variable vector function and \(\xi \) $\xi$ is a sufficiently fast mixing stationary stochastic process. It is known since (Khasminskii in Theory Probab Appl 11:211–228, 1966) that if \({\bar{X}}\) $\overline{X}$ is the averaged motion then \(G^{\varepsilon }={\varepsilon }^{-1/2}(X^{\varepsilon }-{\bar{X}})\) $G^{\epsilon }={\epsilon }^{-1/2}(X^{\epsilon }-\overline{X})$ weakly converges to a Gaussian process G. We will show that for each \({\varepsilon }\) $\epsilon$ the processes \(\xi \) $\xi$ and G can be redefined on a sufficiently rich probability space without changing their distributions so that \(E\sup _{0\le t\le T}|G^{\varepsilon }(t)-G(t)|^{2\,M} =O({\varepsilon }^{{\delta }}),\,{\delta }>0\) $E{sup}_{0\le t\le T}{|G^{\epsilon }(t)-G(t)|}^{2M}=O({\epsilon }^{\delta }),\delta >0$ which gives also \(O({\varepsilon }^{{\delta }/3})\) $O({\epsilon }^{\delta /3})$ Prokhorov distance estimate between the distributions of \(G^{\varepsilon }\) $G^{\epsilon }$ and G. This provides also convergence estimates in the Kantorovich–Rubinstein (or Wasserstein) metrics. In the product case \(B(x,\xi )={\Sigma }(x)\xi \) $B(x,\xi )=\mathrm{\Sigma }(x)\xi$ we obtain also almost sure convergence estimates of the form \(\sup _{0\le t\le T}|G^{\varepsilon }(t)-G(t)| =O({\varepsilon }^{\delta })\) ${sup}_{0\le t\le T}|G^{\epsilon }(t)-G(t)|=O({\epsilon }^{\delta })$ a.s., as well as the Strassen’s form of the law of iterated logarithm for \(G^{\varepsilon }\) $G^{\epsilon }$ . We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems.