Some strong limit theorems in averaging

Abstract

The paper deals with the fast-slow motions setups in the discrete time $X^{\hat{I}}\mu ((n+1)\hat{I}\mu )=X^{\hat{I}}\mu (n\hat{I}\mu )+\hat{I}\mu B(X^{\hat{I}}\mu (n\hat{I}\mu ),\hat{I}¾(n))$, $n=0,1,...,[T/\hat{I}\mu ]$ and the continuous time $\frac{d{X}^{\hat{I}}\mu (t)}{dt}=B(X^{\hat{I}}\mu (t),\hat{I}¾(t/\hat{I}\mu )).t\in [0,T]$ where $B$ is a smooth vector function and $\hat{I}¾$ is a sufficiently fast mixing stationary stochastic process. It is known since 1966 (Khasminskii) that if $\bar{X}$ is the averaged motion then $G^{\hat{I}}\mu =\hat{I}{\mu }^{-1/2}(X^{\hat{I}}\mu -\bar{X})$ weakly converges to a Gaussian process $G$. We will show that for each $\hat{I}\mu$ the processes $\hat{I}¾$ and $G$ can be redefined on a sufficiently rich probability space without changing their distributions so that $E{\sup }_{0\le t\le T}|G^{\hat{I}}\mu (t)-G(t){|}^{2M}=O(\hat{I}{\mu }^{\hat{I}}´)$, $\hat{I}´>0$ which gives also $O(\hat{I}{\mu }^{\hat{I}´/3})$ Prokhorov distance estimate between the distributions of $G^{\hat{I}}\mu$ and $G$. In the product case $B(x,\hat{I}¾)=\hat{I}\pounds (x)\hat{I}¾$ we obtain almost sure convergence estimates of the form ${\sup }_{0\le t\le T}|G^{\hat{I}}\mu (t)-G(t)|=O(\hat{I}{\mu }^{\hat{I}}´)$ a.s., as well as the functional form of the law of iterated logarithm for $G^{\hat{I}}\mu$. We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems.

arXiv admin note: text overlap with arXiv:2105.01940, arXiv:2111.05390