Iterated Ergodic Theorems and Erd\" os--R\' enyi law of large numbers

TL;DR

This paper extends classical ergodic theorems to iterated sums and integrals of vector processes and establishes an Erdős–Rényi law of large numbers for these iterated structures, broadening the scope of ergodic theory.

Contribution

It introduces ergodic theorems for multiple iterated sums and integrals of vector processes and proves an Erdős–Rényi law for these iterated forms, which is a novel extension.

Findings

Ergodic theorems for iterated sums and integrals of vector processes.

A version of the Erdős–Rényi law of large numbers for iterated sums and integrals.

Abstract

We obtain ergodic theorems for multiple iterated sums and integrals of the form $\Sigma^{(\nu)}(t)=\sum_{0\leq k_1<...<k_\nu\leq t}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and $\Sigma^{(\nu)}(t)=\int_{0\leq s_1\leq...\leq s_\nu\leq t}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$ where $\{\xi(k)\}_{-\infty<k<\infty}$ and $\{\xi(s)\}_{-\infty<s<\infty}$ are vector processes for which standard ergodic theorems, i.e. when $\nu=1$, hold true. At the end we prove also a version of the Erd\" os--R\" enyi law of large numbers for iterated sums and integrals.