  We consider again the fast-slow motions setups in the continuous time $\frac
{dX_N(t)}{dt}=N^{1/2} \sig(X_N(t))(\xi(tN))+b(X_N(t)),\, t\in [0,T]$ and the
discrete time
$X_N((n+1)/N)=X_N(n/N)+N^{-1/2}\sig(X_N(n/N))\xi(n)+N^{-1}b(X_N(n/N)),\,
n=0,1,...,[TN]$ where $\sig$ and $b$ are smooth matrix and vector functions,
respectively, $\xi$ is a centered vector stationary stochastic process with
weak dependence in time and $N$ is a big parameter. We obtain estimates for the
almost
  sure approximations of the process $X_N$ by certain diffusion process $\Sig$.
In \cite{FK} and in other recent
  papers concerning similar setups the results were obtained relying fully on
the rough paths theory. Here we derive
  our probabilistic results as corollaries of quite general deterministic
estimates which are obtained with all details
  provided following somewhat ideology of the rough paths theory but not
relying on this theory per se
  which should allow a more general readership to follow complete arguments.
