On the First and the Second Borel-Cantelli Lemmas
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916081466081280 |
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| author | Xie, Jian-Sheng Wang, Qihang |
| author_facet | Xie, Jian-Sheng Wang, Qihang |
| contents | Let $\{A_n\}_{n=1}^\infty$ be a sequence of events and let $\displaystyle S:=\sum_{n=1}^\infty 1_{A_n}$. We present in this note equivalent characterizations for the statements $\mathbb{P} (S<\infty)=1$ and $\mathbb{P} (S=\infty)=1$ respectively. These characterizations are of Borel-Cantelli lemma type and of Kochen-Stone lemma type respectively, which could be regarded as the most general version of the first and the second Borel-Cantelli Lemmas. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_02725 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the First and the Second Borel-Cantelli Lemmas Xie, Jian-Sheng Wang, Qihang Probability 60A05, 60A10, 60F15 Let $\{A_n\}_{n=1}^\infty$ be a sequence of events and let $\displaystyle S:=\sum_{n=1}^\infty 1_{A_n}$. We present in this note equivalent characterizations for the statements $\mathbb{P} (S<\infty)=1$ and $\mathbb{P} (S=\infty)=1$ respectively. These characterizations are of Borel-Cantelli lemma type and of Kochen-Stone lemma type respectively, which could be regarded as the most general version of the first and the second Borel-Cantelli Lemmas. |
| title | On the First and the Second Borel-Cantelli Lemmas |
| topic | Probability 60A05, 60A10, 60F15 |
| url | https://arxiv.org/abs/2401.02725 |