Is the Kalman Filter a Best Linear Unbiased Estimator (BLUE) for Heteroscedastic Noise? According to the Gauss-Markov Theorem, a ordinary least squares estimator is BLUE if the noise entering a system is uncorrelated with zero mean and is homoscedastic (has a constant finite variance). I am aware that a Kalman Filter applied to a system with additive noise of known mean and variance but non-gaussian distribution is  BLUE. Does this imply that the noise has to be homoscedastic ? Or does the KF have a trick up its sleeve ?
Kalman filter is the best linear estimator regardless of stationarity or Gaussianity. Also in the Gaussian case it does not require stationarity (unlike Wiener filter). In the linear Gaussian case Kalman filter is also a MMSE estimator or the conditional mean.