A mathematical preprint reports a way to put a ceiling on the chance that a dynamical system enters an unsafe region during a fixed time horizon, even when only limited moment information — summary statistics describing its starting uncertainty — is known. The question is the worst case across all initial probability measures compatible with that information, not the outcome for one selected starting distribution.
Under the paper’s stated Assumption 1 and feasibility conditions, the measure relaxation is reported to have the same optimal value as the original worst-case probability problem. That is a conditional mathematical result: it depends on the assumptions being met.
How the ceiling is built
At the center of the method is an optimization over probability measures, mathematical descriptions of how probability is spread across possible states. The main program maximizes the mass of an auxiliary measure on the unsafe set while requiring that it be dominated by a terminal measure.
A companion functional formulation is interpreted as a generalized probabilistic barrier-certificate problem — a search for a mathematical condition in which moment information shapes the initial-state constraint.
To make the problem computable, Lasserre’s moment-SOS hierarchy turns it into a sequence of finite convex optimization programs. Each relaxation level supplies an upper bound, and under mild assumptions the sequence converges monotonically.
The framework can also treat the moments as uncertain within box-bounded ranges rather than fixed values. The paper says this keeps the problem convex and produces more conservative bounds.
What the model examples showed
In the academic example, the upper-bound sequence stabilized around 0.31 at relaxation order d = 4. A Monte Carlo check for one admissible initial distribution estimated a probability of 0.0794, below the optimized worst-case value.
The orbital demonstration needed a Taylor expansion of the dynamics to fit the polynomial-program framework. The authors warn that this approximation can introduce significant error over extended horizons.
For a fixed unsafe region in the orbital model, the bound seemed to stall around 0.6183 at d = 7, with solving time around 51.43 seconds. In a separate comparison at d = 8, the reported upper bounds were 0.7364 for the region labeled K2 and 0.5198 for K3; the authors considered the K3 bound potentially loose.
With moment ambiguity included, the orbital bound stabilized after d = 7 at around 0.9089 for κ = 0.008. For a moving unsafe region, the bound plateaued at approximately 0.72 after d = 7, with 47.3 seconds of computation; the estimated critical time was 0.7180.
Why the figures need caution
The paper’s caveats matter: these are finite-order upper bounds, and the hierarchy’s convergence statement depends on its assumptions. In the orbital example, the Taylor approximation may introduce significant error over extended horizons, while the K3 result was explicitly described as potentially loose.
Taken together, the work offers a way to bound a modeled worst-case outcome when initial uncertainty is only partly known. Its interpretation remains tied to the chosen model, the available moment information and the relaxation level used in the calculation.
Paper data and sources
Original title: Worst-Case Probability Bounds for Finite-Horizon Safety under Moment Uncertainty
Authors: Renato Loureiro, Torbjørn Cunis
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text