Anti-concentration of random variables from zero-free regions
| Autor: | Michelen, Marcus, Sahasrabudhe, Julian |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Discrete Analysis 2022:13 |
| Druh dokumentu: | Working Paper |
| Popis: | This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\{0,\ldots,n\}$ with $\mathbb{P}(X = 0)\mathbb{P}(X = n) > 0$ and with probability generating function $f_X$. We show that if all of the zeros $\zeta$ of $f_X$ satisfy $|\arg(\zeta)| \geq \delta$ and $R^{-1} \leq |\zeta| \leq R$ then \[ \operatorname{Var}(X) \geq c R^{-2\pi/\delta}n, \] where $c > 0$ is a absolute constant. We show that this result is sharp, up to the factor $2$ in the exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord type theorem for random variables that are not necessarily sums of i.i.d.\ random variables. Comment: 29 pages |
| Databáze: | arXiv |
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