Bernstein-type and Bennett-type inequalities for unbounded matrix martingales
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913700167811072 |
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| author | Kroshnin, Alexey Suvorikova, Alexandra |
| author_facet | Kroshnin, Alexey Suvorikova, Alexandra |
| contents | We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued martingale processes with unbounded observations from the Hermitian space $\mathbb{H}(d)$. Specifically, we assume that the $ψ_α$-Orlicz (quasi-)norms of their difference process are bounded for some $α> 0$. Further, we generalize the obtained result by replacing the ambient dimension $d$ with the effective rank of the covariance of the observations. To illustrate the applicability of the results, we prove several corollaries, including an empirical version of Bernstein's inequality and an extension of the bounded difference inequality, also known as McDiarmid's inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bernstein-type and Bennett-type inequalities for unbounded matrix martingales Kroshnin, Alexey Suvorikova, Alexandra Probability Statistics Theory 60E15 We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued martingale processes with unbounded observations from the Hermitian space $\mathbb{H}(d)$. Specifically, we assume that the $ψ_α$-Orlicz (quasi-)norms of their difference process are bounded for some $α> 0$. Further, we generalize the obtained result by replacing the ambient dimension $d$ with the effective rank of the covariance of the observations. To illustrate the applicability of the results, we prove several corollaries, including an empirical version of Bernstein's inequality and an extension of the bounded difference inequality, also known as McDiarmid's inequality. |
| title | Bernstein-type and Bennett-type inequalities for unbounded matrix martingales |
| topic | Probability Statistics Theory 60E15 |
| url | https://arxiv.org/abs/2411.07878 |