In a linear Kalman filter, we assume that the state and measurement noise are white noise N(0,Q) and N(0,R) respectively. Is it common practice to test these hypothesis? And what are the most common statistical tests used in this case?
The linear Kalman filter still minimises the mean squared error of the estimate even if noise are non-Gaussian white, it's just that in the Gaussian case you will have the full posterior distribution from the mean and covariance estimates alone.
So in practice it wouldn't be absolutely necessary to test for Gaussian noise, and have the Kalman filter still 'work'.