This note presents the box Gaussian mixture filter (BGMF), which is an efficient filter for the systems with mainly linear measurements but enables utilizing highly nonlinear measurements. BGMF contains a new way to approximate the prior distributions with a Gaussian mixture, whose components have small covariances. In this note we present results on the weak convergence of BGMF. In simulations, we see that in our hybrid position example BGMF outperforms the conventional particle filter.