Kernel based Dirichlet sequences

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2021-05-31 DOI:10.3150/22-bej1500
P. Berti, E. Dreassi, F. Leisen, L. Pratelli, P. Rigo
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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.
基于核的狄利克雷序列
设$X=(X_1,X_2,\ldots)$是一个随机变量序列,其值在标准空间$(S,\mathcal{B})$中。假设\ begin{collecte*}X_1\sim\nu\quad\text{and}\quad P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)=\frac{θ\nu(\cdot)+\sum_{i=1}^nK(X_i)(\cdot)}$\mathcal{B}$。然后,只要$K$是$\nu$的正则条件分布,给定$\mathcal{B}$的任何子$\sigma$字段,$X$是可交换的。在此假设下,$X$具有经典Dirichlet序列的所有主要性质,包括Sethuraman表示、共轭性质和预测分布的全变分收敛性。如果$\mu$是经验测度的弱极限,则提供了$\mu$a.s.离散、a.s.非原子或$\mu\ll\nu$a.s.的条件。还证明了两个CLT。第一个涉及稳定收敛,而第二个涉及总变差距离。
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来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
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. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
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