{"title":"Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions","authors":"Quan Shi, Matthias Winkel","doi":"10.30757/alea.v20-25","DOIUrl":null,"url":null,"abstract":"Forman et al. (2020+) constructed $(\\alpha,\\theta)$-interval partition evolutions for $\\alpha\\in(0,1)$ and $\\theta\\ge 0$, in which the total sums of interval lengths (\"total mass\") evolve as squared Bessel processes of dimension $2\\theta$, where $\\theta\\ge 0$ acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson--Dirichlet interval partitions. In this paper we study symmetry properties of $(\\alpha,\\theta)$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${\\rm SSIP}^{(\\alpha)}(\\theta_1,\\theta_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $\\theta_1\\ge 0$ and $\\theta_2\\ge 0$. They also have squared Bessel total mass processes of dimension $2\\theta$, where $\\theta=\\theta_1+\\theta_2-\\alpha\\ge-\\alpha$ covers emigration as well as immigration. Under the constraint $\\max\\{\\theta_1,\\theta_2\\}\\ge\\alpha$, we prove that an ${\\rm SSIP}^{(\\alpha)}(\\theta_1,\\theta_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $\\alpha$ and $\\theta$, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2020-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","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.v20-25","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: 3
Abstract
Forman et al. (2020+) constructed $(\alpha,\theta)$-interval partition evolutions for $\alpha\in(0,1)$ and $\theta\ge 0$, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension $2\theta$, where $\theta\ge 0$ acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson--Dirichlet interval partitions. In this paper we study symmetry properties of $(\alpha,\theta)$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${\rm SSIP}^{(\alpha)}(\theta_1,\theta_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $\theta_1\ge 0$ and $\theta_2\ge 0$. They also have squared Bessel total mass processes of dimension $2\theta$, where $\theta=\theta_1+\theta_2-\alpha\ge-\alpha$ covers emigration as well as immigration. Under the constraint $\max\{\theta_1,\theta_2\}\ge\alpha$, we prove that an ${\rm SSIP}^{(\alpha)}(\theta_1,\theta_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $\alpha$ and $\theta$, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.
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.