L1-DRAC: Distributionally Robust Adaptive Control — Global Results

The paper’s Theorem 4.1 is proved by (i) combining two prior bounds via the triangle inequality in Wasserstein distance and (ii) deriving high-probability bounds by first controlling L^{2p}-moments on the original probability space using Minkowski’s inequality, then applying Markov’s inequality, under the design conditions (21) and (24) and constructions in Lemmas 4.4 and 4.8. This is explicitly shown in the manuscript’s statement and proof of Theorem 4.1 and the supporting lemmas. By contrast, the model’s Step 4 applies Markov’s inequality to an optimal transport coupling that realizes W_{2p}, thereby obtaining a tail bound under the coupling measure rather than under the original law of the processes. That does not establish the claimed pathwise (probability) bounds for the actual processes. The rest of the model’s steps (triangle inequality for distributional bounds, and design-condition verification) align with the paper’s logic. See Theorem 4.1 and its proof and the use of Lemmas 4.4 and 4.8 for the paper’s correct approach, and note the role of conditions (21) and (24) in enabling those lemmas.