### Abstract

For a discrete random variable, what information can we deduce from the roots of its probability generating function? We consider a sequence of random variables X_n taking values between 0 and n, and let P_n(z) be its probability generating function. We show that if none of the complex zeros of the polynomials P_n(z) are contained in a neighborhood of 1 in the complex plane then a central limit theorem occurs, provided the variance of X_n isn't subpolynomial in n. This result is sharp a sense that will be made precise, and thus disproves a conjecture of Pemantle and improves upon various results in the literature. This immediately improves a multivariate central limit theorem of Ghosh, Liggett and Pemantle, and has ramifications for certain variables that arise in graph theory contexts. Time permitting, we will describe other of our results connecting the location of the zeros of P_n(z) and the distribution of the random variables X_n. This is based on joint work with Julian Sahasrabudhe.