  Ensemble filters implement sequential Bayesian estimation by representing the
probability distribution by an ensemble mean and covariance. Unbiased square
root ensemble filters use deterministic algorithms to produce an analysis
(posterior) ensemble with prescribed mean and covariance, consistent with the
Kalman update. This includes several filters used in practice, such as the
Ensemble Transform Kalman Filter (ETKF), the Ensemble Adjustment Kalman Filter
(EAKF), and a filter by Whitaker and Hamill. We show that at every time index,
as the number of ensemble members increases to infinity, the mean and
covariance of an unbiased ensemble square root filter converge to those of the
Kalman filter, in the case a linear model and an initial distribution of which
all moments exist. The convergence is in $L^{p}$ and the convergence rate does
not depend on the model dimension. The result holds in the infinitely
dimensional Hilbert space as well.
