Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting

The paper’s Theorem 2.4 proves the L2-error bound (E|μ_J^{EK,N}[φ] − μ_J[φ]|^2)^{1/2} ≤ C(N^{-1/2}+ε) for test functions φ of class P2, under linear observations h and dynamics Ψ that are sup-norm ε-close to an affine map, with Ψ Lipschitz and Gaussian initial data; the proof proceeds by a triangle inequality splitting finite-ensemble fluctuation (1/√N) and mean-field bias (O(ε)), relying on moment bounds and Lipschitz stability of the P, Q, Bj, and mean-field EnKF maps . The candidate solution mirrors this structure exactly: (i) uniform high-moment bounds; (ii) Lipschitz continuity of operators in a weighted BL/Wasserstein framework; (iii) 1/√N convergence to the mean-field limit; (iv) O(ε) bias via continuity around an affine anchor and exactness of the mean-field EnKF in the linear-Gaussian case . Differences are largely notational (their BL2 metric vs the paper’s dg) and stylistic (they argue a uniform normalization lower bound for Bj), but they entail the same logical steps and assumptions as the paper’s proof sketch and appendices.