  Consider the class of Ensemble Square Root filtering algorithms for the
numerical approximation of the posterior distribution of nonlinear Markovian
signals partially observed with linear observations corrupted with independent
measurement noise. We analyze the asymptotic behavior of these algorithms in
the large ensemble limit both in discrete and continuous time. We identify
limiting mean-field processes on the level of the ensemble members, prove
corresponding propagation of chaos results and derive associated convergence
rates in terms of the ensemble size. In continuous time we also identify the
stochastic partial differential equation driving the distribution of the
mean-field process and perform a comparison with the Kushner-Stratonovich
equation.
