Under a squeezing condition on the dynamics, a discrete-time square-root ensemble Kalman filter (and its surrogate-model variant) achieves long-time mean state estimation error of order ε, the observation noise level, plus surrogate error δ.
Mourtada, Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices, The Annals of Statistics, 50 (2022), pp
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Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems
Under a squeezing condition on the dynamics, a discrete-time square-root ensemble Kalman filter (and its surrogate-model variant) achieves long-time mean state estimation error of order ε, the observation noise level, plus surrogate error δ.