Random valuations
Abstract
A valuation is a finitely additive function on the family of compact convex sets in ℝd. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, σ-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs (F,r), where F is a non-empty closed convex set and r>0, with each pair contributing r1F∩ K=∅. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.
Related
BibTeX
@misc{ilienko2026random,
title = {{Random valuations}},
author = {Andrii Ilienko and Ilya Molchanov and Tommaso Visonà},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.19976},
eprint = {2608.19976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19976},
}
RIS
TY - UNPB
TI - Random valuations
AU - Andrii Ilienko
AU - Ilya Molchanov
AU - Tommaso Visonà
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2608.19976
UR - https://arxiv.org/abs/2608.19976
AB - A valuation is a finitely additive function on the family of compact convex sets in ℝ^d. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, σ-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs (F,r), where F is a non-empty closed convex set and r>0, with each pair contributing r1_F∩ K=∅. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.
ER -