When Kalman's assumptions are relaxed From Kalman's seminal paper "A New Approach to Linear Filtering and Prediction Problem", it is clear that Kalman's exposition is based on the following fundamental assumptions:


Measurements that are linearly related to the state.
Measurements are corrupted by white noise:
2a: Serially uncorrelated noise.
2b: Zero mean noise.
Measurements follow the Gaussian distribution.


When these assumptions are met, the Kalman Filter is "the best filter of any conceivable form", according to Maybeck (1979) "Stochastic Models, Estimation, and Control, Vol 1", page 7.

I am looking for an overview of sources that deal with the effect on the estimation error of relaxing the stated assumptions, and I am hoping someone here can help me.
HI: I haven't read it but this discusses the non-linear case so it may be helpful atleast to get an idea of what the issues are. http://sites.utexas.edu/renato/files/2017/04/recursiveUpdate_v3.pdf