一个标记和定律决定随机质量划分与$\Xi$-coalescents
One marked-sum law determines random mass partitions and $Ξ$-coalescents
- University of Bologna（博洛尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明一个实值标记和定律可完全决定随机质量划分及$\Xi$-coalescent的碰撞测度，并给出构造与不可区分性结果。
AI中文摘要：
设$P=(P_j)$为一个随机质量划分，$(X_j)$为独立于$P$的独立同分布实值标记。我们构造一个固定的标记分布，使得实值随机变量$\sum_j P_j X_j$的分布决定$P$在Kingman单纯形上的分布。该观测映射是一个仿射拓扑嵌入。一种构造使用具有有理无关指标的稳定定律的无限卷积；对于每个给定的$q\in(0,\infty)$，可以改为选择对称、中心化、复合泊松且属于$L^q$的确定标记。在任意已知正时刻，从单例开始的$\Xi$-coalescent的排序块频率的分布——等价地，该时刻其所有有限限制的可交换划分概率函数——决定有限碰撞测度$\Xi$。因此，组合这两个结果即可从一个实值标记和的分布识别$\Xi$。在$\Lambda$子类中，非对称Bernoulli着色决定归一化碰撞测度，任何中心化非常数标记（其某阶绝对矩$a\in(1,4)$有限而所有更高阶绝对矩无限）也是如此。我们还证明，对于每个$d\ge 2$，支撑在严格递减的$(d+1)$元正质量组（具有固定总质量）上的两个不同分布，对于任何支撑在至多$d$个点上的标记仍然不可区分。对于每个固定类型数$d\ge 2$，这一障碍产生不同的归一化碰撞测度，它们具有相同的无突变中性$d$型$\Xi$-Fleming–Viot转移半群。
英文摘要：
Let $P=(P_j)$ be a random mass partition and let $(X_j)$ be iid real marks, independent of $P$. We construct a fixed mark distribution for which the law of the real random variable $\sum_j P_j X_j$ determines the law of $P$ on the Kingman simplex. The observation map is an affine topological embedding. One construction uses an infinite convolution of stable laws with rationally independent indices; for every prescribed $q\in(0,\infty)$, a determining mark can instead be chosen symmetric, centered, compound Poisson, and in $L^q$. At any known positive time, the law of the ranked block frequencies of a $Ξ$-coalescent started from singletons$\unicode{x2014}$equivalently, the exchangeable partition probability functions of all its finite restrictions at that time$\unicode{x2014}$determines the finite collision measure $Ξ$. Composing the two results therefore identifies $Ξ$ from the law of one real marked sum. Within the $Λ$ subclass, an asymmetric Bernoulli coloring determines the normalized collision measure, as does any centered nonconstant mark whose absolute moment of some order $a\in(1,4)$ is finite while every higher absolute moment is infinite. We also prove that, for every $d\ge 2$, a pair of distinct laws supported on strictly decreasing $(d+1)$-tuples of positive masses with a fixed total mass remains indistinguishable for every mark supported on at most $d$ points. For every fixed number $d\ge 2$ of types, this obstruction yields distinct normalized collision measures with the same mutation-free neutral $d$-type $Ξ$-Fleming$\unicode{x2013}$Viot transition semigroup.
