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 $Σ^{(ν)}(t)=\sum_{0\leq k_1<...<k_ν\leq t}ξ(k_1)\otimes\cdots\otimesξ(k_ν)$, $t\in[0,T]$ and $Σ^{(ν)}(t)=\int_{0\leq s_1\leq...\leq s_ν\leq t}ξ(s_1)\otimes\cdots\otimesξ(s_ν)ds_1\cdots ds_ν$ where $\{ξ(k)\}_{-\infty<k<\infty}$ and $\{ξ(s)\}_{-\infty<s<\infty}$ are vector processes for which standard ergodic theorems, i.e. when $ν=1$, 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.