Kac 多项式和导数的孔半径

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY Stochastic Processes and their Applications Pub Date : 2024-05-25 DOI:10.1016/j.spa.2024.104386

Hoi H. Nguyen , Oanh Nguyen

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摘要

方差为 1 的独立系数 Kac 多项式是研究最多的随机多项式模型之一。

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Hole radii for the Kac polynomials and derivatives

The Kac polynomial $f_{n} (x) = \sum_{i = 0}^{n} ξ_{i} x^{i}$ with independent coefficients of variance 1 is one of the most studied models of random polynomials.

It is well-known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work (Michelen, 2020), Michelen showed that the holes at $\pm 1$ are of order $1 / n$ . We show that in fact, all the hole radii are of the same order. The same phenomenon is established for the derivatives of the Kac polynomial as well.

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来源期刊

Stochastic Processes and their Applications 数学-统计学与概率论

CiteScore

2.90

自引率

7.10%

发文量

180

审稿时长

23.6 weeks

期刊介绍： Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests. Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.