Probabilistic Limit Theorems Induced by the Zeros of Polynomials

Nils Heerten; Holger Sambale; Christoph Thäle

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arXiv · 2207.08469

Probabilistic Limit Theorems Induced by the Zeros of Polynomials

Nils Heerten·Holger Sambale·Christoph Thäle

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Abstract

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.

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BibTeX RIS

Nils Heerten, Holger Sambale, Christoph Thäle. 2023-12-01. Probabilistic Limit Theorems Induced by the Zeros of Polynomials. https://arxiv.org/abs/2207.08469

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