AI 中文总结
研究随机划分中分量计数的二阶极限定理，在次临界和临界状态下分别得到平稳Ornstein-Uhlenbeck高斯过程和平稳M/M/∞排队过程作为极限。
AI 中文摘要
在最近的一篇论文中，Banderier等人（2024）研究了由参数为$\alpha\in(0,1)$和$\theta>-\alpha$的中国餐馆过程诱导的随机划分的分量计数的极限行为。设$C_j(n)$表示$\{1,\ldots,n\}$的划分中大小为$j$的分量个数，并考虑$j=j_n\to\infty$当$n\to\infty$。他们揭示了$C_{j_n}(n)$的一阶极限行为中的相变，其中临界状态对应于$j_n\sim rn^{\alpha/(1+\alpha)}$，$r>0$。一个自然的后续问题是理解相应的二阶波动。我们建立了计数过程$(C_{j_n}(n(1+t/j_n)_+))_{t\in\mathbb R}$在次临界状态（$j_n\ll n^{\alpha/(1+\alpha)}$）和临界状态下的二阶极限定理。在次临界状态下，经过适当归一化后，极限是一个平稳的Ornstein-Uhlenbeck高斯过程，而在临界状态下，极限是一个平稳的$M/M/\infty$排队过程。我们还在临界状态下建立了更精细的点过程收敛。实际上，我们为更一般的Karlin无限瓮模型建立了二阶极限定理，然后将分析应用于中国餐馆过程。
英文摘要
In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameters $α\in(0,1)$ and $θ>-α$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\{1,\ldots,n\}$ and consider $j=j_n\to\infty$ as $n\to\infty$. They identified a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\sim rn^{α/(1+α)}$ for some $r>0$. A natural next question is to understand the corresponding second-order fluctuations. We establish second-order limit theorems in the critical regime and, under an additional rate condition in the subcritical regime ($j_n\ll n^{α/(1+α)}/(\log\log n)^{1/(1+α)}$), for the counting process $(C_{j_n}(n(1+t/j_n)_+))_{t\in\mathbb R}$. In the subcritical regime, after appropriate normalization, the limit is a stationary Ornstein--Uhlenbeck Gaussian process, whereas in the critical regime the limit is a stationary $M/M/\infty$ queue. We also establish a more refined point-process convergence in the critical regime. We first establish these results for the more general Karlin infinite urn model and then adapt the analysis to the Chinese restaurant process. For the latter model, most of our limit theorems are established in the quenched sense.
Comments54 pages; major revision. Several mistakes in the previous version have been corrected, and a couple improvements have been made. The convergence in Proposition 1.4 (now 1.3) has been improved to almost sure convergence. The previous Corollary 4.2 has been replaced by a convergence of measure-valued process in D space in Theorem 4.2
