关于由断棍生成的随机环境中嵌套占位方案的中间层：重尾情形

The Bernoulli sieve: an overview

The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.

Small parts in the Bernoulli sieve

Sampling from a random discrete distribution induced by a `stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline.

随机环境中占用方案的新近性及其在tries中的应用

Asymptotics of the occupancy scheme in a random environment and its applications to tries

本文用嵌套盒子序列的占用方案变体，研究有限状态马尔可夫链多个副本轨迹首次分叉时间的新近性，并推广至随机转移概率及随机/非随机场景下观察全部轨迹的时间。

Limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals $n$, we prove several functional limit theorems (FLTs) in the Skorohod space $D[0,1]$ endowed with the $J_{1}$- or $M_{1}$-topology for the number $K_{n}^{*}(t)$ of boxes containing at most $[n^{t}]$ balls, $t\in[0,1]$, and the random distribution function $K_{n}^{*}(t)/K_{n}^{*}(1)$, as $n\to\infty$. The limit processes for $K_{n}^{*}(t)$ are of the form $(X(1)-X((1-t)-))_{t\in[0,1]}$, where $X$ is either a Brownian motion, a spec…

嵌套复Wishart极值的单侧乘积尺度上界

One-Sided Product-Scale Upper Bounds for Nested Complex Wishart Extremes

本文针对嵌套复Wishart极值的最大特征值软边缘，证明了其上、下尾联合概率的有限N单侧乘积尺度上界，推导了分离稀疏网格的矩估计，适用于超临界分离场景。

有色top-$m$-to-random洗牌的显式截止轮廓

Explicit cutoff profiles for colored top-$m$-to-random shuffles

研究花环积$C_p\wr S_n$上的$p$色top-$m$-to-random洗牌，利用Nakano-Sadahiro-Sakurai基元素$B_m$得到精确的嵌套集占用混合，将似然比简化为单一统计量$L_p$，从而导出分离度、$L^\infty(U)$、全变差、$L^q(U)$、$\chi^2$和相对熵的精确公式，并给出截止轮廓。