Operation with Concentration Inequalities
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908801149435904 |
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| author | Louart, Cosme |
| author_facet | Louart, Cosme |
| contents | Following the concentration of the measure theory formalism, we consider the transformation $Φ(Z)$ of a random variable $Z$ having a general concentration function $α$. If the transformation $Φ$ is $λ$-Lipschitz with $λ>0$ deterministic, the concentration function of $Φ(Z)$ is immediately deduced to be equal to $α(\cdot/λ)$. If the variations of $Φ$ are bounded by a random variable $Λ$ having a concentration function (around $0$) $β: \mathbb R_+\to \mathbb R$, this paper sets that $Φ(Z)$ has a concentration function analogous to the so-called parallel product of $α$ and $β$. With this result at hand (i) we express the concentration of random vectors with independent heavy-tailed entries, (ii) given a transformation $Φ$ with bounded $k^{\text{th}}$ differential, we express the so-called ``multilevel'' concentration of $Φ(Z)$ as a function of $α$, and the operator norms of the successive differentials up to the $k^{\text{th}}$ (iii) we obtain a heavy-tailed version of the Hanson--Wright inequality.
Finally, in order to rigorously handle the algebraic operations that arise on concentration functions (parallel sums, parallel products, and non-unique pseudo-inverses), we develop at the beginning of the paper a functional framework based on maximally monotone set-valued operators, which provides a natural and coherent formalism for studying these transformations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operation with Concentration Inequalities Louart, Cosme Probability Functional Analysis 60E15, 46A55, 60G70, 47H05, 49J53 Following the concentration of the measure theory formalism, we consider the transformation $Φ(Z)$ of a random variable $Z$ having a general concentration function $α$. If the transformation $Φ$ is $λ$-Lipschitz with $λ>0$ deterministic, the concentration function of $Φ(Z)$ is immediately deduced to be equal to $α(\cdot/λ)$. If the variations of $Φ$ are bounded by a random variable $Λ$ having a concentration function (around $0$) $β: \mathbb R_+\to \mathbb R$, this paper sets that $Φ(Z)$ has a concentration function analogous to the so-called parallel product of $α$ and $β$. With this result at hand (i) we express the concentration of random vectors with independent heavy-tailed entries, (ii) given a transformation $Φ$ with bounded $k^{\text{th}}$ differential, we express the so-called ``multilevel'' concentration of $Φ(Z)$ as a function of $α$, and the operator norms of the successive differentials up to the $k^{\text{th}}$ (iii) we obtain a heavy-tailed version of the Hanson--Wright inequality. Finally, in order to rigorously handle the algebraic operations that arise on concentration functions (parallel sums, parallel products, and non-unique pseudo-inverses), we develop at the beginning of the paper a functional framework based on maximally monotone set-valued operators, which provides a natural and coherent formalism for studying these transformations. |
| title | Operation with Concentration Inequalities |
| topic | Probability Functional Analysis 60E15, 46A55, 60G70, 47H05, 49J53 |
| url | https://arxiv.org/abs/2402.08206 |