Some strong limit theorems in averaging

TL;DR

This paper establishes strong limit theorems for averaging in fast-slow stochastic systems, providing convergence rates and probabilistic estimates for the difference between the scaled process and its Gaussian limit.

Contribution

It introduces new probabilistic bounds and almost sure convergence results for the averaging principle in systems with fast mixing stochastic processes.

Findings

Weak convergence of scaled deviations to Gaussian process

Explicit convergence rate estimates in L^p norm

Almost sure convergence and law of iterated logarithm results

Abstract

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