arXiv 1809.092762018-09math.PR

Pitman $α$ -多样性的Berry-Esseen定理

A Berry-Esseen theorem for Pitman's $α$-diversity

Emanuele Dolera, Stefano Favaro

DOI：: 10.1214/19-AAP1518

摘要：

本文研究了由两参数Poisson-Dirichlet分布驱动的可交换随机变量的随机样本$(X_1, \dots, X_n)$中不同元素个数的随机变量$K_n$，$PD(α,θ)$ .对于$α\in(0,1)$，\cite{Pit(06)}中的定理3.8表明$\frac{K_n}{n^α}\stackrel{\text{a.s.}}{\longrightarrow} S_{α,θ}$为$n\rightarrow+\infty$。这里，$S_{α,θ}$是根据所谓的标度Mittag-Leffler分布进行分布的随机变量。我们的主要结果表明，$$ \sup_{x \geq 0} \Big| \ppsf\Big[\frac{K_n}{n^α} \leq x \Big] - \ppsf[S_{α,θ} \leq x] \Big| \leq \frac{C(α, θ)}{n^α} $$持有一个明确的常数$C(α, θ)$。证明的关键成分是一种新的概率表示$K_n$作为复合分布和新的，完善的版本的某些定量界的泊松近似和复合泊松分布。

英文摘要：

This paper is concerned with the study of the random variable $K_n$ denoting the number of distinct elements in a random sample $(X_1, \dots, X_n)$ of exchangeable random variables driven by the two parameter Poisson-Dirichlet distribution, $PD(α,θ)$. For $α\in(0,1)$, Theorem 3.8 in \cite{Pit(06)} shows that $\frac{K_n}{n^α}\stackrel{\text{a.s.}}{\longrightarrow} S_{α,θ}$ as $n\rightarrow+\infty$. Here, $S_{α,θ}$ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states that $$ \sup_{x \geq 0} \Big| \ppsf\Big[\frac{K_n}{n^α} \leq x \Big] - \ppsf[S_{α,θ} \leq x] \Big| \leq \frac{C(α, θ)}{n^α} $$ holds with an explicit constant $C(α, θ)$. The key ingredients of the proof are a novel probabilistic representation of $K_n$ as compound distribution and new, refined versions of certain quantitative bounds for the Poisson approximation and the compound Poisson distribution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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