P. Berti, E. Dreassi, F. Leisen, L. Pratelli, P. Rigo
{"title":"Kernel based Dirichlet sequences","authors":"P. Berti, E. Dreassi, F. Leisen, L. Pratelli, P. Rigo","doi":"10.3150/22-bej1500","DOIUrl":null,"url":null,"abstract":"Let $X=(X_1,X_2,\\ldots)$ be a sequence of random variables with values in a standard space $(S,\\mathcal{B})$. Suppose \\begin{gather*} X_1\\sim\\nu\\quad\\text{and}\\quad P\\bigl(X_{n+1}\\in\\cdot\\mid X_1,\\ldots,X_n\\bigr)=\\frac{\\theta\\nu(\\cdot)+\\sum_{i=1}^nK(X_i)(\\cdot)}{n+\\theta}\\quad\\quad\\text{a.s.} \\end{gather*} where $\\theta>0$ is a constant, $\\nu$ a probability measure on $\\mathcal{B}$, and $K$ a random probability measure on $\\mathcal{B}$. Then, $X$ is exchangeable whenever $K$ is a regular conditional distribution for $\\nu$ given any sub-$\\sigma$-field of $\\mathcal{B}$. Under this assumption, $X$ enjoys all the main properties of classical Dirichlet sequences, including Sethuraman's representation, conjugacy property, and convergence in total variation of predictive distributions. If $\\mu$ is the weak limit of the empirical measures, conditions for $\\mu$ to be a.s. discrete, or a.s. non-atomic, or $\\mu\\ll\\nu$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bernoulli","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3150/22-bej1500","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 6
Abstract
Let $X=(X_1,X_2,\ldots)$ be a sequence of random variables with values in a standard space $(S,\mathcal{B})$. Suppose \begin{gather*} X_1\sim\nu\quad\text{and}\quad P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)=\frac{\theta\nu(\cdot)+\sum_{i=1}^nK(X_i)(\cdot)}{n+\theta}\quad\quad\text{a.s.} \end{gather*} where $\theta>0$ is a constant, $\nu$ a probability measure on $\mathcal{B}$, and $K$ a random probability measure on $\mathcal{B}$. Then, $X$ is exchangeable whenever $K$ is a regular conditional distribution for $\nu$ given any sub-$\sigma$-field of $\mathcal{B}$. Under this assumption, $X$ enjoys all the main properties of classical Dirichlet sequences, including Sethuraman's representation, conjugacy property, and convergence in total variation of predictive distributions. If $\mu$ is the weak limit of the empirical measures, conditions for $\mu$ to be a.s. discrete, or a.s. non-atomic, or $\mu\ll\nu$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance.
期刊介绍:
BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work.
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