Erd\"os-R\'enyi law of large numbers in the averaging setup

TL;DR

This paper extends the Erdős-Rényi law of large numbers to averaging setups involving stochastic processes and dynamical systems, under conditions of fast mixing and large deviations, in both discrete and continuous time.

Contribution

It generalizes the Erdős-Rényi law to new settings involving fast mixing processes and dynamical systems with large deviations, including continuous-time flows.

Findings

Erdős-Rényi law holds for fast mixing stochastic processes.

Results apply to flows with suspension representations.

Applicable in both discrete and continuous time cases.

Abstract

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.