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

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.