Kernel-Based Prediction of Non-Markovian Time Series

The paper’s Proposition 2.1 states and sketches a proof of the bound E_ν[(K^*_{μ_N} N_{μ,N} E_{L,N}[Y|·] − E[Y|·])^2] = O(L/N, (log N)^{p_d/2} N^{-1/(2d)}, L^{-2β/d}), decomposing variance, discretization, and bias terms; the sketch appeals to a Monte Carlo bound for coefficients and a recent eigenfunction convergence rate on compact manifolds to obtain the middle dimension-dependent term, plus Weyl-law bias for Sobolev F (see the statement and sketch in the paper’s Section 2.1, Proposition 2.1, and the surrounding text ). The candidate solution reaches the same overall rate using a different route: an operator/spectral-projector perturbation (Davis–Kahan) plus concentration for empirical kernel operators and a Sobolev/Weyl-law bias. That proof is plausible under its added assumptions (notably that the kernel is a spectral multiplier of the Laplace–Beltrami operator), and it delivers a sharper O(N^{-1/2}) subspace term which indeed implies the paper’s stated O((log N)^{p_d/2} N^{-1/(2d)}) bound. However, the model’s argument glosses over data-dependence in the coefficient term and implicitly replaces the paper’s graph-Laplacian/Markov-operator discretization with an empirical integral-operator viewpoint; still, as a proof sketch it supports the same rate with a different mechanism. Net: both arguments support the same bound; the paper’s route is manifold-spectral, the model’s route is integral-operator perturbative.