# Anti-concentration of random variables from zero-free regions

@inproceedings{Michelen2021AnticoncentrationOR, title={Anti-concentration of random variables from zero-free regions}, author={Marcus Michelen and Julian Sahasrabudhe}, year={2021} }

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 fX . We show that if all of the zeros ζ of fX satisfy | arg(ζ)| > δ and R 6 |ζ| 6 R then Var(X) > cRn, 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…

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