An Erd\" os-R\' enyi law for nonconventional sums

TL;DR

This paper establishes an Erdős-Rényi type law of large numbers for nonconventional sums involving i.i.d. variables and a bounded Borel function, extending classical results to more complex sum structures.

Contribution

It introduces a new Erdős-Rényi law for sums of the form involving multiple indices, based on nonconventional large deviations, which broadens the scope of classical probability laws.

Findings

Proves an Erdős-Rényi law for nonconventional sums.

Utilizes nonconventional large deviations techniques.

Extends classical laws to sums with multiple index dependencies.

Abstract

We obtain rge Erd\" os-R\' enyi type law of large numbers for "nonconventional" sums of the form $S_n=\sum^n_{m=1}F(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of i.i.d. random variables and $F$ is a bounded Borel function. The proof relies on nonconventional large deviations obtained in [8].