ANTI-CONCENTRATION OF RANDOM VARIABLES FROM ZERO-FREE REGIONS

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, ... , n} with P(X = 0)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)| >= delta and R-1 <= |zeta|<= R then Var(X) >= cR(-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.