{"title":"Tree convolution for probability distributions\nwith unbounded support","authors":"E. Davis, David Jekel, Zhichao Wang","doi":"10.30757/alea.v18-58","DOIUrl":null,"url":null,"abstract":"We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in\"An operad of non-commutative independences defined by trees\"(Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $\\mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $\\boxplus_{\\mathcal{T}}(\\mu_1,\\dots,\\mu_N)$ for arbitrary probability measures $\\mu_1$, ..., $\\mu_N$ on $\\mathbb{R}$ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from doi:10.4064/dm797-6-2020. We prove a general limit theorem for iterated $\\mathcal{T}$-free convolution similar to Bercovici and Pata's results in the free case in\"Stable laws and domains of attraction in free probability\"(Annals of Mathematics, 1999, doi:10.2307/121080), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/alea.v18-58","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in"An operad of non-commutative independences defined by trees"(Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $\mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $\boxplus_{\mathcal{T}}(\mu_1,\dots,\mu_N)$ for arbitrary probability measures $\mu_1$, ..., $\mu_N$ on $\mathbb{R}$ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from doi:10.4064/dm797-6-2020. We prove a general limit theorem for iterated $\mathcal{T}$-free convolution similar to Bercovici and Pata's results in the free case in"Stable laws and domains of attraction in free probability"(Annals of Mathematics, 1999, doi:10.2307/121080), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.
我们发展了第二作者和刘卫华在《树定义的非交换独立算子》(数学论文,2020,doi:10.4064/dm797-6-2020)中研究的树卷积的复分析观点,推广了自由卷积、布尔卷积、单调卷积和正交卷积。特别地,对于$N$正则树的每个根子树$\mathcal{T}$(顶点由交替字符串标记),我们定义了任意概率测度$\ma_1$,…的卷积$\boxplus_{\mathcal{T}}(\mu_1,\dots,\mu_N)$$\mu_N$在$\mathbb{R}$上使用Cauchy变换的特定定点方程。卷积运算遵循doi:10.4064/dm797-6-2020中的树运算器的运算器结构。我们证明了迭代$\mathcal{T}$自由卷积的一个一般极限定理,类似于Bercovici和Pata在“自由概率中的稳定定律和吸引域”(Annals of Mathematics,1999,doi:10.2307/12180)中在自由情况下的结果,并推导出每个经典稳定定律在吸引域中的测度的极限定理。
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.