Geodesic
Anti-concentration of random variables from zero-free regions
Discrete analysis (2022) Cet article a éte moissonné depuis la source Scholastica

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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 $ζ$ of $f_X$ satisfy $|\arg(ζ)| \geq δ$ and $R^{-1} \leq |ζ| \leq R$ then \[ \operatorname{Var}(X) \geq c R^{-2π/δ}n, \] 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.
Publié le : 
@article{DAS_2022_a7,
     author = {Marcus Michelen and Julian Sahasrabudhe},
     title = {Anti-concentration of random variables from zero-free regions},
     journal = {Discrete analysis},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2022_a7/}
}
TY  - JOUR
AU  - Marcus Michelen
AU  - Julian Sahasrabudhe
TI  - Anti-concentration of random variables from zero-free regions
JO  - Discrete analysis
PY  - 2022
UR  - http://geodesic.mathdoc.fr/item/DAS_2022_a7/
LA  - en
ID  - DAS_2022_a7
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%0 Journal Article
%A Marcus Michelen
%A Julian Sahasrabudhe
%T Anti-concentration of random variables from zero-free regions
%J Discrete analysis
%D 2022
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%F DAS_2022_a7
Marcus Michelen; Julian Sahasrabudhe. Anti-concentration of random variables from zero-free regions. Discrete analysis (2022). http://geodesic.mathdoc.fr/item/DAS_2022_a7/