Iterated Ergodic Theorems and Erd\" os--R\' enyi law of large numbers
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909694211129344 |
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| author | Kifer, Yuri |
| author_facet | Kifer, Yuri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iterated Ergodic Theorems and Erd\" os--R\' enyi law of large numbers Kifer, Yuri Probability 34C29 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. |
| title | Iterated Ergodic Theorems and Erd\" os--R\' enyi law of large numbers |
| topic | Probability 34C29 |
| url | https://arxiv.org/abs/2501.15633 |