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 ${\Sigma }^{(\nu )}(t)={\sum }_{0\le k_1<...<k_{\nu }\le t}\xi (k_1)\otimes \cdots \otimes \xi (k_{\nu })$, $t\in [0,T]$ and ${\Sigma }^{(\nu )}(t)={\int }_{0\le s_1\le ...\le s_{\nu }\le t}\xi (s_1)\otimes \cdots \otimes \xi (s_{\nu })d{s}_1\cdots d{s}_{\nu }$ 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.

13 pages