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Mathematics > Probability

arXiv:2207.08469 (math)
[Submitted on 18 Jul 2022 (v1), last revised 1 Dec 2023 (this version, v4)]

Title:Probabilistic Limit Theorems Induced by the Zeros of Polynomials

Authors: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.
Comments: Changed in [V2]: Correction of a typo, additional literature added; Changed in [v3]: Thoroughly revised version; Changed in [v4]: additional literature added
Subjects: Probability (math.PR)
MSC classes: 05A16, 52B20, 60C05, 60F05, 60F15
Cite as: arXiv:2207.08469 [math.PR]
  (or arXiv:2207.08469v4 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.2207.08469

Submission history

From: Nils Heerten [view email]
[v1] Mon, 18 Jul 2022 09:44:06 UTC (100 KB)
[v2] Thu, 21 Jul 2022 17:24:43 UTC (100 KB)
[v3] Fri, 3 Mar 2023 11:06:35 UTC (138 KB)
[v4] Fri, 1 Dec 2023 13:09:12 UTC (257 KB)
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