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

We obtain ergodic theorems for multiple iterated sums and integrals of the form ${\mathrm{\Sigma }}^{(\nu )}(t)={\sum }_{0\le k_1<\mathrm{.}\mathrm{.}\mathrm{.}<k_{\nu }\le t}\xi (k_1)\otimes \cdots \otimes \xi (k_{\nu })$, $t\in [0,T]$ and ${\mathrm{\Sigma }}^{(\nu )}(t)={\int }_{0\le s_1\le \mathrm{.}\mathrm{.}\mathrm{.}\le s_{\nu }\le t}\xi (s_1)\otimes \cdots \otimes \xi (s_{\nu })d{s}_1\cdots d{s}_{\nu }$ where $\{\xi (k){\}}_{-\mathrm{\infty }<k<\mathrm{\infty }}$ and $\{\xi (s){\}}_{-\mathrm{\infty }<s<\mathrm{\infty }}$ 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.