Probabilistic reachable sets of stochastic nonlinear systems with contextual uncertainties

The paper proves almost-uniform convergence of the resampling-based chance-constraint approximation and convergence of optimal values/solutions (Theorems 1–2) via (i) Monte Carlo convergence to a resampling-law probability using an existing smooth SAA result plus Egorov, and (ii) convergence of the resampling law to the true law using LS-CDE, Scheffé/Portmanteau, and induction in k. The candidate model reaches the same conclusions with a different proof: a VC–Glivenko–Cantelli step for the Monte Carlo error, plus stability of k-step pushforwards in total variation from L1-consistency of the conditional density estimator, and the same Egorov/Berge-type stability for optimization. Both arguments are substantively correct; each has a mild regularity gap (paper: asserting pointwise-in-ξ convergence from an L2 result; model: requiring uniform-on-compacts L1-consistency in ξ). These gaps are standard and can be patched by clarifying the regularity assumptions.