Anticoncentration of random variables from zerofree regions
Abstract
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 LittlewoodOfford type theorem for random variables that are not necessarily sums of i.i.d.\ random variables.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.07699
 Bibcode:
 2021arXiv210207699M
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics
 EPrint:
 25 pages