论文标题
关于固定环境中的嵌套占用方案的中间级别，破坏了II产生的随机环境
On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking II
论文作者
论文摘要
随机环境中的嵌套占用方案是对随机环境（随机概率）中经典的Karlin无限球占用方案的概括。与盒子集合的Karlin方案不同，有一个箱子的嵌套层次结构，并且盒子的命中概率是根据单位质量的迭代碎片定义的。在本文中，我们假设随机的碎片定律是通过杆折叠给出的，在这种情况下，第一级盒子定义的无限占用方案被称为伯努利筛子。假设$ n $ balls已被投掷，请用$ k_n（j）$表示$ j $ th级别的占用盒子的数量，如果$ j = j_n \ to \ j_n \ to \ infty $和$ j_n = o（\ log n）$ as $ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ infty $。我们证明了向量$（k_n（\ lfloor j_n u_1 \ rfloor）的多维中心限制定理$ j_n = o（\ log n）^{1/2} $。 n）^{1/3}）$进行了分析。
A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that $n$ balls have been thrown, denote by $K_n(j)$ the number of occupied boxes in the $j$th level and call the level $j$ intermediate if $j=j_n\to\infty$ and $j_n=o(\log n)$ as $n\to\infty$. We prove a multidimensional central limit theorem for the vector $(K_n(\lfloor j_n u_1\rfloor),\ldots, K_n(\lfloor j_n u_\ell\rfloor)$, properly normalized and centered, as $n\to\infty$, where $j_n\to\infty$ and $j_n=o((\log n)^{1/2})$. The present paper continues the line of investigation initiated in Buraczewski, Dovgay and Iksanov [Electron. J. Probab. 25: paper no. 123, 2020] in which the occupancy of intermediate levels $j_n\to\infty$, $j_n=o((\log n)^{1/3})$ was analyzed.
