The infimum values of the probability functions for some infinitely divisible distributions motivated by Chvátal's theorem
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913192964259840 |
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| author | Hu, Ze-Chun Lu, Peng Zhou, Qian-Qian Zhou, Xing-Wang |
| author_facet | Hu, Ze-Chun Lu, Peng Zhou, Qian-Qian Zhou, Xing-Wang |
| contents | Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq κE[X])$, where $κ$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07678 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The infimum values of the probability functions for some infinitely divisible distributions motivated by Chvátal's theorem Hu, Ze-Chun Lu, Peng Zhou, Qian-Qian Zhou, Xing-Wang Probability Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq κE[X])$, where $κ$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions. |
| title | The infimum values of the probability functions for some infinitely divisible distributions motivated by Chvátal's theorem |
| topic | Probability |
| url | https://arxiv.org/abs/2308.07678 |