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Sequences of discrete random variables are studied whose probabilitygenerating functions are zero-free in a sector of the complex plane around thepositive real axis. Sharp bounds on the cumulants of all orders are stated,leading to Berry-Esseen bounds, moderate deviation results, concentrationinequalities and mod-Gaussian convergence. In addition, an alternate proof ofthe cumulant bound with improved constants for a class of polynomials all ofwhose roots lie on the unit circle is provided. A variety of examples isdiscussed in detail.

Probabilistic Limit Theorems Induced by the Zeros of Polynomials