Some Strong Limit Theorems in Averaging

The paper deals with the fast-slow motions setups in the discrete time X^ε((n+1)ε)=X^ε(nε)+εB(X^ε(nε),ξ(n)), n=0,1,..,[T/ε] and the continuous time dX^ε(t)dt=B(X^ε(t),ξ(t/ε)),t∈[0,T] where B is a smooth in the first variable vector function and ξ is a sufficiently fast mixing stationary stochastic process. It is known since (Khasminskii in Theory Probab Appl 11:211–228, 1966) that if X¯ is the averaged motion then G^ε=ε^-1/2(X^ε-X¯) weakly converges to a Gaussian process G. We will show that for each ε the processes ξ and G can be redefined on a sufficiently rich probability space without changing their distributions so that Esup_0≤t≤T|Gε(t)-G(t)|^2M=O(ε^δ),δ>0 which gives also O(ε^δ/3) Prokhorov distance estimate between the distributions of G^ε and G. This provides also convergence estimates in the Kantorovich–Rubinstein (or Wasserstein) metrics. In the product case B(x,ξ)=Σ(x)ξ we obtain also almost sure convergence estimates of the form sup_0≤t≤T|G^ε(t)-G(t)|=O(ε^δ) a.s., as well as the Strassen’s form of the law of iterated logarithm for G^ε. We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems.