  We extend the Erd\H os-R\' enyi law of large numbers to the averaging setup
both in discrete and continuous time cases. We consider both stochastic
processes and dynamical systems as fast motions whenever they are fast mixing
and satisfy large deviations estimates. In the continuous time case we consider
flows with large deviations estimates which allow a suspension representation
and it turns out that fast mixing of corresponding base transformation suffices
for our results.
