## Section: New Results

### Sequential data assimilation: ensemble Kalman filter vs. particle filter

Participants : François Le Gland, Valérie Monbet.

Surprisingly, very little was known about the asymptotic behaviour of the ensemble Kalman filter [44] , [45] , [46] , whereas on the other hand, the asymptotic behaviour of many different classes of particle filters is well understood, as the number of particles goes to infinity. Interpreting the ensemble elements as a population of particles with mean–field interactions, and not only as an instrumental device producing an estimation of the hidden state as the ensemble mean value, it has been possible to prove the convergence of the ensemble Kalman filter, with a rate of order $1/\sqrt{N}$, as the number $N$ of ensemble elements increases to infinity [62] . In addition, the limit of the empirical distribution of the ensemble elements has been exhibited, which differs from the usual Bayesian filter. The next step has been to prove (by induction) the asymptotic normality of the estimation error, i.e. to prove a central limit theorem for the ensemble Kalman filter.