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Optimality and numerical efficiency are well known properties of the Kalman filter, whereas its stability property, though equally classical and important in practice, is less often mentioned in the recent literature. The stability of the Kalman filter is usually ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. Such classical results cannot be applied to output error systems, in which the process noise is totally absent. It is shown in this paper that the uniform complete observability is sufficient to ensure the stability of the Kalman filter applied to time varying output error systems, regardless of the stability of the considered system itself.