Boolean cumulants and subordination in free probability

IF 0.9 4区数学 Q4 PHYSICS, MATHEMATICAL

Random Matrices-Theory and Applications Pub Date : 2019-07-26 DOI:10.1142/S2010326321500362

F. Lehner, K. Szpojankowski

引用次数: 2

Abstract

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

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自由概率中的布尔累积量与隶属关系

隶属性是自由加性和乘法卷积解析方法的基础。我们将此方法扩展到更一般的设置，并证明了自由随机变量的条件期望[公式:见文本]和Borel函数[公式:见文本]是一个解决方案。这个结果允许显式地计算非交换多项式的分布，其形式为[公式:见文本]。主要工具是一个新的布尔累积量条件期望的组合公式和一个相应的解的条件期望的解析公式，推广了加性和乘性自由卷积的从属公式。在最后一节中，我们用一步一步的显式计算和所有必要成分的阐述来说明结果。

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.