有界周长核分拆统计量的渐近正态性与集中不等式
Asymptotic Normality and Concentration Inequalities of Statistics of Core Partitions with Bounded Perimeters
AI总结：
本文证明了有界周长均匀随机核分拆的若干统计量在Kolmogorov和Wasserstein距离下具有渐近正态性与次高斯性，并借助Stein方法、Hoeffding分解及Efron-Stein不等式等工具，将Zaleski猜想推广到严格$(n,dn+1)$-核分拆情形。
摘要：
自Anderson（2002）关于互素s,t的$(s,t)$-核分拆数量的工作以来，核分拆受到了广泛关注。近年来，研究随机同时核分拆大小的极限分布的兴趣日益增长。本文在Kolmogorov距离和Wasserstein $W_1$距离下证明了具有有界周长的均匀随机核分拆的某些统计量的渐近正态性，包括随机（严格）$n$-核分拆的长度和大小、Durfee正方形的长度以及随机自共轭$n$-核分拆的大小。据此，我们证明了这些统计量是次高斯的。这与Even-Zohar（2022）研究的互素s,t的随机$(s, t)$-核分拆大小的渐近行为形成对比，后者依分布收敛于Watson的$U^2$分布。我们的结果表明，当$d \ge 3$固定且$n$趋于无穷时，随机严格$(n, dn+1)$-核分拆大小的分布是渐近正态的，这是Zaleski猜想（2017）的一个类比，并涵盖了Komlós、Sergel和Tusnády的结果（2020）作为特例。我们的证明综合了多种组合与概率工具，包括基于Hoeffding分解的Stein方法、Hoeffding组合中心极限定理、乘积空间和切片上的Efron-Stein不等式以及Pólya频率序列的渐近分析。此外，我们的方法可能适用于研究具有某些全局依赖结构且可分解为适当混合形式的随机变量泛函的渐近正态性。
英文摘要：
Core partitions have attracted much attention since Anderson's work (2002) on the number of $(s,t)$-core partitions for coprime $s,t$. Recently, there has been a growing interest in studying the limiting distributions of the sizes of random simultaneous core partitions. In this paper, we prove the asymptotic normality of certain statistics of uniform random core partitions with bounded perimeters in the Kolmogorov and Wasserstein $W_1$ distances, including the length and size of a random (strict) $n$-core partition, the length of the Durfee square and the size of a random self-conjugate $n$-core partition. Accordingly, we prove that these statistics are subgaussian. This contrasts with the asymptotic behavior of the size of a random $(s, t)$-core partition for coprime $s,t$ studied by Even-Zohar (2022), which converges in law to Watson's $U^2$ distribution. Our results show that the distribution of the size of a random strict $(n, dn+1)$-core partition is asymptotically normal when $d \ge 3$ is fixed and $n$ tends to infinity, which is an analog of Zaleski's conjecture (2017) and covers Komlós, Sergel, and Tusnády's result (2020) as a special case. Our proof integrates a variety of combinatorial and probabilistic tools, including Stein's method based on Hoeffding decomposition, Hoeffding's combinatorial central limit theorem, the Efron-Stein inequalities on product spaces and slices, and asymptotics of Pólya frequency sequences. Furthermore, our approach is potentially applicable to the study of the asymptotic normality of functionals of random variables with certain global dependence structures that can be decomposed into appropriate mixture forms.
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