The Boolean quadratic forms and tangent law

IF 0.9 4区数学 Q4 PHYSICS, MATHEMATICAL

Random Matrices-Theory and Applications Pub Date : 2024-01-29 DOI:10.1142/s2010326324500047

Wiktor Ejsmont, Patrycja Hęćka

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引用次数: 0

Abstract

In [W. Ejsmont and F. Lehner, The free tangent law, Adv. Appl. Math. 121 (2020) 102093], we study the limit sums of free commutators and anticommutators and show that the generalized tangent function $\frac{tan z}{1 - x tan z}$ describes the limit distribution. This is the generating function of the higher order tangent numbers of Carlitz and Scoville (see (1.6) in [L. Carlitz and R. Scoville, Tangent numbers and operators, Duke Math. J. 39 (1972) 413–429]) which arose in connection with the enumeration of certain permutations. In this paper, we continue to study the limit of weighted sums of Boolean commutators and anticommutators and we show that the shifted generalized tangent function appears in a limit theorem. In order to do this, we shall provide an arbitrary cumulants formula of the quadratic form. We also apply this result to obtain several results in a Boolean probability theory.

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布尔二次型和正切定律

In [W. Ejsmont and F. Lehner, The free tangent law, Adv.Ejsmont and F. Lehner, The free tangent law, Adv.Appl.121 (2020) 102093]中，我们研究了自由换元和反换元的极限和，并证明广义正切函数 tanz1-xtanz 描述了极限分布。这就是 Carlitz 和 Scoville 的高阶正切数的生成函数（见 [L. Carlitz and R. Scoville] 中的 (1.6) 。Carlitz and R. Scoville, Tangent numbers and operators, Duke Math.39 (1972) 413-429]中的 (1.6)），它是在枚举某些排列时产生的。在本文中，我们将继续研究布尔换元数和反换元数的加权和的极限，并证明移项广义切线函数出现在极限定理中。为此，我们将提供二次型的任意累积公式。我们还将应用这一结果来获得布尔概率论中的若干结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Random Matrices-Theory and Applications Decision Sciences-Statistics, Probability and Uncertainty

CiteScore

1.90

自引率

11.10%

发文量

期刊介绍： Random Matrix Theory (RMT) has a long and rich history and has, especially in recent years, shown to have important applications in many diverse areas of mathematics, science, and engineering. The scope of RMT and its applications include the areas of classical analysis, probability theory, statistical analysis of big data, as well as connections to graph theory, number theory, representation theory, and many areas of mathematical physics. Applications of Random Matrix Theory continue to present themselves and new applications are welcome in this journal. Some examples are orthogonal polynomial theory, free probability, integrable systems, growth models, wireless communications, signal processing, numerical computing, complex networks, economics, statistical mechanics, and quantum theory. Special issues devoted to single topic of current interest will also be considered and published in this journal.