One marked-sum law determines random mass partitions and Ξ-coalescents
Abstract
Let P=(Pj) be a random mass partition and let (Xj) be iid real marks, independent of P. We construct a fixed mark distribution for which the law of the real random variable ∑j Pj Xj 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∈(0,∞), a determining mark can instead be chosen symmetric, centered, compound Poisson, and in Lq. At any known positive time, the law of the ranked block frequencies of a Ξ-coalescent started from singletonsunicodex2014equivalently, the exchangeable partition probability functions of all its finite restrictions at that timeunicodex2014determines 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∈(1,4) is finite while every higher absolute moment is infinite. We also prove that, for every d≥ 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≥ 2 of types, this obstruction yields distinct normalized collision measures with the same mutation-free neutral d-type Ξ-Flemingunicodex2013Viot transition semigroup.
Versions
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BibTeX
@misc{lenzi2026markedsum,
title = {{One marked-sum law determines random mass partitions and $Ξ$-coalescents}},
author = {Jacopo Lenzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12790},
eprint = {2609.12790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12790},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - One marked-sum law determines random mass partitions and Ξ-coalescents
AU - Jacopo Lenzi
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12790
UR - https://arxiv.org/abs/2609.12790
AB - 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 ∑_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∈(0,∞), 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 singletonsunicodex2014equivalently, the exchangeable partition probability functions of all its finite restrictions at that timeunicodex2014determines 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∈(1,4) is finite while every higher absolute moment is infinite. We also prove that, for every d≥ 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≥ 2 of types, this obstruction yields distinct normalized collision measures with the same mutation-free neutral d-type Ξ-Flemingunicodex2013Viot transition semigroup.
ER -