Hereditary Hsu-Robbins-Erdös Law of Large Numbers
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915966363893760 |
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| author | Berkes, Istvan Karatzas, Ioannis Schachermayer, Walter |
| author_facet | Berkes, Istvan Karatzas, Ioannis Schachermayer, Walter |
| contents | We show that every sequence $f_1, f_2, \cdots$ of real-valued random variables with $\sup_{n \in \N} \E (f_n^2) < \infty$ contains a subsequence $f_{k_1}, f_{k_2}, \cdots$ converging in \textsc{Cesàro} mean to some $\,f_\infty \in \mathbb{L}^2$ {\it completely,} to wit, $ \sum_{N \in \N} \, ¶\left( \bigg| \frac{1}{N} \sum_{n=1}^N f_{k_n} - f_\infty \bigg| > \eps \right)< \infty\,, \quad \forall ~ \eps > 0\,; $ and {\it hereditarily,} i.e., along all further subsequences as well. We also identify a condition, slightly weaker than boundedness in $ \mathbb{L}^2,$ which turns out to be not only sufficient for the above hereditary complete convergence in \textsc{Cesàro} mean, but necessary as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hereditary Hsu-Robbins-Erdös Law of Large Numbers Berkes, Istvan Karatzas, Ioannis Schachermayer, Walter Probability Primary 60A10, 60F15, Secondary 60G42, 60G46 G.3 We show that every sequence $f_1, f_2, \cdots$ of real-valued random variables with $\sup_{n \in \N} \E (f_n^2) < \infty$ contains a subsequence $f_{k_1}, f_{k_2}, \cdots$ converging in \textsc{Cesàro} mean to some $\,f_\infty \in \mathbb{L}^2$ {\it completely,} to wit, $ \sum_{N \in \N} \, ¶\left( \bigg| \frac{1}{N} \sum_{n=1}^N f_{k_n} - f_\infty \bigg| > \eps \right)< \infty\,, \quad \forall ~ \eps > 0\,; $ and {\it hereditarily,} i.e., along all further subsequences as well. We also identify a condition, slightly weaker than boundedness in $ \mathbb{L}^2,$ which turns out to be not only sufficient for the above hereditary complete convergence in \textsc{Cesàro} mean, but necessary as well. |
| title | Hereditary Hsu-Robbins-Erdös Law of Large Numbers |
| topic | Probability Primary 60A10, 60F15, Secondary 60G42, 60G46 G.3 |
| url | https://arxiv.org/abs/2503.19484 |