Decoupling Maximal Inequalities
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929433120604160 |
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| author | Kontorovich, Aryeh |
| author_facet | Kontorovich, Aryeh |
| contents | A {\em maximal inequality} seeks to estimate $\mathbb{E}\max_i X_i$ in terms of properties of the $X_i$. When the latter are independent, the union bound (in its various guises) can yield tight upper bounds. If, however, the $X_i$ are strongly dependent, the estimates provided by the union bound will be rather loose. In this note, we show that for non-negative random variables, pairwise independence suffices for the maximal inequality to behave comparably to its independent version. The condition of pairwise independence may be relaxed to a kind of negative dependence, and even the latter admits violations -- provided these are properly quantified. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14150 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Decoupling Maximal Inequalities Kontorovich, Aryeh Probability A {\em maximal inequality} seeks to estimate $\mathbb{E}\max_i X_i$ in terms of properties of the $X_i$. When the latter are independent, the union bound (in its various guises) can yield tight upper bounds. If, however, the $X_i$ are strongly dependent, the estimates provided by the union bound will be rather loose. In this note, we show that for non-negative random variables, pairwise independence suffices for the maximal inequality to behave comparably to its independent version. The condition of pairwise independence may be relaxed to a kind of negative dependence, and even the latter admits violations -- provided these are properly quantified. |
| title | Decoupling Maximal Inequalities |
| topic | Probability |
| url | https://arxiv.org/abs/2302.14150 |