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.
Categories: math.PR math.CO