On the nonexistence of event-based triggers that preserve Gaussian state in presence of package-drop

State estimation is a core objective in cyber-physical systems. In the state estimation problem over linear systems, the Kalman filter is the standard solution. The filter is the format on which the solutions to subsequent estimation problems are based. Among these problems are the estimation problem in the presence of packet drops and estimation problem involving event-based triggers. We study in this paper both phenomena simultaneously. In an attempt to find the Kalman-like filter, which proves the Gaussianity of the state and offers a set of update equations, our paper shows that no such filter exists. More precisely, one cannot find an event-based trigger such that under possible packet drops, the state variable remains a Gaussian variable. This conclusion can be reasonably extended to a more general setting.