ERROR BOUNDS OF THE INVARIANT STATISTICS IN MACHINE LEARNING OF ERGODIC ITÔ DIFFUSIONS

The paper’s Proposition 3.1 (one-point statistics) and Proposition 3.2 (two-point statistics) are stated and proved using (i) a one-step O(ε) bound (Lemma 3.1) feeding into a perturbation bound for geometrically ergodic chains (Proposition 2.1) and (ii) a linear-response/resolvent argument under Assumptions 2.5–2.8 (Theorem 2.2) together with a spectral gap (Assumption 2.6) to control the derivative and obtain a 1/(1−λ) prefactor; see the statements and proof sketches in the paper for Prop. 3.1 and Prop. 3.2, including Eqs. (32)–(33) for the two-point result . The candidate solution reproduces these bounds via a telescoping identity for Markov operators plus one-step stability and the same linear-response/resolvent machinery for the two-point statistic; the decomposition of k_ε−k and the spectral-gap-based resolvent estimate match the paper closely. Minor differences are stylistic or involve slightly stronger intermediate assumptions (e.g., a uniform-in-ε derivative bound for ∂P_ε^t in the one-step diffusion bound and an omitted explicit δ factor in the EM one-step display), but they do not affect the final conclusions, which coincide with the paper’s results.