One marked-sum law determines random mass partitions and $Ξ$-coalescents

Abstract

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.