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Probabilistic limit theorems induced by the zeros of polynomials

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  • Nils Heerten
  • Holger Sambale
  • Christoph Thäle

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.

Suggested Citation

  • Nils Heerten & Holger Sambale & Christoph Thäle, 2024. "Probabilistic limit theorems induced by the zeros of polynomials," Mathematische Nachrichten, Wiley Blackwell, vol. 297(5), pages 1772-1792, May.
    • Handle: RePEc:bla:mathna:v:297:y:2024:i:5:p:1772-1792
    DOI: 10.1002/mana.202300109
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    References listed on IDEAS

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    1. Zikai Wu & Hongxia Du, 2014. "Some limit properties for the coefficients of the q-Catalan numbers," Indian Journal of Pure and Applied Mathematics, Springer, vol. 45(4), pages 469-478, August.
    2. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
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