Topology, Convergence, and Reconstruction of Predictive States

The paper’s Theorem 5 states the estimator as Ĉ(←−X−→X)(Ĉ(←−X←−X) − ζ I)^{-1}|δ_{←−X}〉 with a rate O((Lζ)^{-1/2} + ζ^{1/2} + γ^{−`} + h_{←−X}(`)) under stationarity and ergodicity. However, two issues arise directly from the text: (i) the regularization sign is written as “−ζ I” both in the KBR statement and in Theorem 5, whereas the standard and necessary Tikhonov regularization is “+ζ I” to ensure invertibility and stability; see their own KBR-like statement immediately preceding Theorem 5 where the same “−ζ I” appears, contradicting standard CME/KBR formulations (and the intended rate expression) . (ii) The tail term is given as γ^{−`}, but the paper has already fixed 0 < γ < 1 in defining the discounted sequence metric and kernel, making γ^{−`} explode as ` increases and thus incompatible with the claimed convergence as ` → ∞; earlier finite-length embedding results in the same paper scale as O(γ^{`}), not γ^{−`} . Moreover, the theorem asserts an almost-sure root-L sampling rate under mere stationarity and ergodicity, which is not justified; obtaining such a rate typically requires mixing assumptions beyond ergodicity. The candidate solution corrects the regularization sign to “+ζ I”, replaces the tail by O(γ^{`}), and explicitly supplies a mixing assumption to justify the root-L behavior, while giving a standard three-term CME error decomposition. These corrections align with standard CME/KBR results and with the paper’s own earlier constructions, yielding a coherent, correct proof and rate.