On the anti-concentration functions of some familiar families of distributions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916190668980224 |
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| author | Hu, Ze-Chun Song, Renming Tan, Yuan |
| author_facet | Hu, Ze-Chun Song, Renming Tan, Yuan |
| contents | Let $\{X_α\}$ be a family of random variables following a certain type of distributions with finite expectation $\mathbf{E}[X_α]$ and finite variance ${\rm Var}(X_α)$, where $α$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \infty)\ni y\to \inf_α\mathbf{P}\left(|X_α-\mathbf{E}[X_α]|\geq y \sqrt{{\rm Var}(X_α)}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, exponential distributions, non-degenerate Gaussian distributions and student's $t$-distribution, the anti-concentration function is not identically zero, while for some other familiar families of distributions, including binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the anti-concentration functions of some familiar families of distributions Hu, Ze-Chun Song, Renming Tan, Yuan Probability Let $\{X_α\}$ be a family of random variables following a certain type of distributions with finite expectation $\mathbf{E}[X_α]$ and finite variance ${\rm Var}(X_α)$, where $α$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \infty)\ni y\to \inf_α\mathbf{P}\left(|X_α-\mathbf{E}[X_α]|\geq y \sqrt{{\rm Var}(X_α)}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, exponential distributions, non-degenerate Gaussian distributions and student's $t$-distribution, the anti-concentration function is not identically zero, while for some other familiar families of distributions, including binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero. |
| title | On the anti-concentration functions of some familiar families of distributions |
| topic | Probability |
| url | https://arxiv.org/abs/2401.09998 |