Central limit theorems from the roots of probability generating functions

For each n, let X-n is an element of {0, ..., n} be a random variable with mean mu(n), standard deviation sigma(n), and let P-n(z) = Sigma P-n (k=0) (X-n = k)z(k), be its probability generating function. We show that if none of the complex zeros of the polynomials {P-n (z)} is contained in a neighborhood of 1 is an element of C and sigma(n) > n(epsilon) for some epsilon > 0, then X-n* = (X-n - mu(n))sigma(-1)(n) is asymptotically normal as n -> infinity: that is, it tends in distribution to a random variable Z similar to N(0, 1). On the other hand, we show that there exist sequences of random variables {X-n} with sigma(n) > C log n for which P-n (z) has no roots near 1 and X-n* is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of P-n(z) and the distribution of the random variable X-n. (C) 2019 Published by Elsevier Inc.