2022/07/18 by Heerten, Nils, Sambale, Holger, Thäle, Christoph
#05A16 #52B20 #60C05 #60F05 #60F15 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2207.08469
Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to Berry-Esseen bounds, moderate deviation results, concentration inequalities and mod-Gaussian convergence. In addition, an alternate proof of the cumulant bound with improved constants for a class of polynomials all of whose roots lie on the unit circle is provided. A variety of examples is discussed in detail.