Local limit theorems for occupancy models

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.

Functional central limit theorems for occupancies and missing mass process in infinite urn models

We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labelled 1,2,... so that the urn $j$ at every draw gets a ball with probability $p_j$, $\sum_j p_j=1$. We prove functional central limit theorems for discrete time and the poissonised version for the urn occupancies process, for the odd-occupancy and for the missing mass processes extending the known non-functional central limit theorems.

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.

A Berry-Esseen bound for the uniform multinomial occupancy model

The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that $$ \sup_{z \in \mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left( \frac{1+(\frac{n}{m})^3}{σ_{n,m}} \right) \quad \mbox{for all $n \ge d$ and $m \ge 2$,} $$ where $W_{n,m}$ and $σ_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variab…

多集合分配占据分布的极限理论：通过MAO范数实现的正态与泊松近似

Limit Theory of the Multi-set Allocation Occupancy (MAO) Distribution: Normal and Poisson Approximations via MAO Norm

本文通过MAO范数研究多集合分配占据分布的极限理论，推导出正态与泊松近似，并验证其在不同渐进行为下的有效性。

Discretized normal approximation by Stein's method

We prove a general theorem to bound the total variation distance between the distribution of an integer valued random variable of interest and an appropriate discretized normal distribution. We apply the theorem to 2-runs in a sequence of i.i.d. Bernoulli random variables, the number of vertices with a given degree in the Erdös-Rényi random graph, and the uniform multinomial occupancy model.

随机轨迹中表观分形维数的有限尺寸占据标度

Finite-size occupancy scaling of apparent fractal dimensions in stochastic trajectories

针对有限随机轨迹中分形维数估计的有限尺寸标度问题，提出基于球箱占据模型的偏差校正方法，通过归一化局部斜率坍塌实现跨模型类的误差降低。

关于Ewens-Pitman模型的中心极限定理

On central limit theorems for Ewens-Pitman model

本文基于中国餐馆过程研究了Ewens-Pitman模型中组件数量的中心极限定理，揭示了波动由采样效应和随机渐近频率波动两部分组成，且在极限条件下两者在给定α-多样性下条件独立。

Stein couplings for normal approximation

In this article we propose a general framework for normal approximation using Stein's method. We introduce the new concept of Stein couplings and we show that it lies at the heart of popular approaches such as the local approach, exchangeable pairs, size biasing and many other approaches. We prove several theorems with which normal approximation for the Wasserstein and Kolmogorov metrics becomes routine once a Stein coupling is found. To illustrate the versatility of our framework we give applications in Hoeffding's combinatorial central limit theorem, functionals in the classic occupancy sche…