Functional Erd\"os-R\'enyi law of large numbers for nonconventional sums under weak dependence

TL;DR

This paper establishes a functional Erdős-Rényi law of large numbers for nonconventional sums involving dependent random vectors with exponential mixing, extending previous results from i.i.d. cases to dependent sequences.

Contribution

It extends the Erdős-Rényi law of large numbers to nonconventional sums of dependent random vectors with weak dependence conditions.

Findings

Proves a functional Erdős-Rényi law for exponentially mixing sequences.

Generalizes previous i.i.d. results to dependent data.

Provides a framework for nonconventional sums under weak dependence.

Abstract

We obtain a functional Erd\H os-R\' enyi law of large numbers for "nonconventional" sums of the form $\Sig_n=\sum_{m=1}^nF(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of exponentially fast $\psi$-mixing random vectors and $F$ is a Borel vector function extendin in several directions our previous result concerning i.i.d. random variables $X_1,X_2,...$.