Many sensor data fusion approaches are based on physically motivated models. Some of them include non-Gaussian noise like radar clutter or GNSS multipath. Classical fusion algorithms like Kalman filters assume Gaussian noise processes. Nevertheless, they are widely applied to state estimation problems under non-Gaussian noise conditions. We investigate in this paper how a unimodal measurement noise distribution affects the distribution of the state vector of a linear Kalman filter. The analysis is based on cumulants which reflect the non-Gaussian character of a distribution. The theoretical results are, that the distribution of the state vector gets closer the Gaussian distribution over time during the settlement process. The higher cumulants converge faster than the variance due to higher exponents. The distribution converges to the Gaussian distribution only if there is no system noise present. Otherwise, there will be a remaining deviation of the converged distribution from the Gaussian one. In a simulation, the theoretical results are proven. The higher cumulants decrease over time until they converge not necessarily to zero. Filters which have to deal with high system noise or only few observations may benefit from an improved model of tracking uncertainty based on the cumulants.
