Mean field limit of Ensemble Square Root filters - discrete and continuous time

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