Ergodicity bounds in the Sliced Wasserstein distance for Schur stable autoregressive processes

The paper’s Theorem 3.8 states three non‑asymptotic bounds for W_p(X_t(x),X_∞). The lower bound and the first upper bound are supported by a correct coupling/mixing argument, yielding W_p(X_t(x),X_∞) ≤ E|Q^t(x−X_∞)| for all p ≥ 1 (see the proof line “Wp(Xt(x),X∞) ≤ ∫ Wp(Xt(x),Xt(y))P(X∞∈dy) ≤ … = E|Qt(x−X∞)|” and display (3.13) in the uploaded PDF ; cf. the statement of Theorem 3.8 ). By contrast, the candidate solution incorrectly asserts that this expectation‑bound only holds for p=1 and must be replaced by an L^p bound for p>1; this is strictly weaker than the paper’s bound and thus incorrect as a critique. For the second upper bound, however, the paper’s proof appears to rely on the step (E|∑_{j≥t+1}Q^jΣξ_j|^p)^{1/p} ≤ (∑_{j≥t+1}E|Q^jΣξ_j|^p)^{1/p}, which is not valid in general under the paper’s minimal hypotheses. This step underpins the sharper tail factor ∥Q∥_*^{t+1}/(1−∥Q∥_*^p)^{1/p} shown in (3.13) (see the detailed chain displayed under Theorem 3.8 ). Without additional structure (e.g., zero‑mean terms enabling von Bahr–Esseen for 1≤p≤2, or a Rosenthal‑type bound), the rigorous generic consequence of Minkowski is only the ℓ^1‑sum bound, leading to the weaker denominator 1−∥Q∥_* that the candidate derives. Thus: the paper’s first and lower bounds check out, the candidate’s criticism of the first upper bound is incorrect, but the paper’s second upper bound needs an extra hypothesis or a revised proof; the candidate provides a safe but weaker alternative.