Fluctuations and Long-Time Stability of Multivariate Ensemble Kalman Filters

We develop a self contained stochastic perturbation theory for discrete generation and multivariate Ensemble Kalman filters. Unlike their continuous-time counterparts, discrete EnKF algorithms are defined through a two steps prediction update mechanism and exhibit non Gaussian fluctuations, even in linear settings. In the multivariate case, these fluctuations take the form of non central Wishart type perturbations, which significantly complicate the mathematical analysis. We establish non asymptotic, time-uniform stability and error estimates for the ensemble covariance matrix processes under minimal structural assumptions on the signal observation model, allowing for possibly unstable dynamics. Our results quantify the impact of ensemble size, dimension, and observation noise, and provide explicit bounds on the propagation of stochastic errors over long time horizons. The analysis relies on a detailed study of stochastic Riccati difference equations driven by matrix-valued noncentral Wishart fluctuations. Beyond their relevance to data assimilation, these results contribute to the probabilistic understanding of ensemble-based filtering methods in high dimension and offer new tools for the analysis of interacting particle systems with matrix-valued dynamics.