An arXiv preprint sets out exactly which jump patterns are compatible with independent increments—a condition in which changes recorded over separate parts of the process do not depend on one another—in a random rule for valuing convex shapes. More formally, it gives an if-and-only-if characterization for non-negative, infinitely divisible, monotone and σ-continuous random valuations.
Here, a valuation is treated as a stochastic process on the space of convex bodies: a rule that assigns values to shapes. The document is an arXiv version 1 preprint dated 20 August 2026, and no empirical sample is reported.
The jump structure
The central construction uses a Lévy measure to describe jumps through pairs made up of a non-empty closed convex set and a positive jump size. Each pair is associated with an intersection indicator, a yes-or-no contribution tied to an intersection involving the shape being valued.
Within the stated assumptions, independent increments are equivalent to obtaining the Lévy measure from a Borel measure on closed convex sets and positive jump sizes through that intersection map. The same class can then be represented in distribution as a deterministic monotone, σ-continuous valuation plus a Poisson point-process sum—a random accumulation of weighted intersection indicators.
The general representation also gives a precise monotonicity rule: the deterministic part must be monotone, and the Lévy measure must be supported on monotone valuations. When the random realizations are almost surely σ-continuous, the Lévy measure is likewise supported on σ-continuous valuations.
A geometric picture under symmetry
For stationary valuations, the representation uses measures on Grassmannians of different dimensions—mathematical spaces associated with subspaces—alongside the positive jump-size axis. In the stationary isotropic case, the one-dimensional Lévy measure is a non-negative sum of jump-size measures weighted by valuation terms V_m(K).
An additional stability condition under dilation of the argument narrows the stationary class further. It restricts the exponent to an integer order determined by the ambient dimension and places the Poisson atoms on affine subspaces of complementary dimension.
The paper also constructs independent non-negative components weakly equivalent to the original stationary isotropic valuation: one component tied to non-empty sets, and others stable under argument dilation at the corresponding orders. That equivalence is only at the level of one-dimensional distributions, so it does not generally mean equality of the joint behavior of several valuation values.
A boundary on the result
The framework has a negative implication too: not every deterministic monotone σ-continuous valuation can arise as the expectation of an infinitely divisible valuation with independent increments.
These are conditional mathematical statements, not empirical estimates. Their scope depends on the assumptions imposed in each result, including non-negativity, infinite divisibility, monotonicity, σ-continuity, stationarity or isotropy, independent increments and, in some cases, a zero deterministic part.
The text also carries a technical warning: the abstract and a later main theorem use conflicting versions of the intersection indicator—empty versus non-empty—so the exact formula needs to be checked against the source before the representation is used.
Paper data and sources
Original title: Random valuations
Authors: Andrii Ilienko, Ilya Molchanov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text