Whiteness hypothesis in Kalman filtering

Question

In Kalman filter mathematical treatment I have always read that a foundamental hypothesis is represented by the whiteness of the process noise. I have tried to do again the mathematical steps in the Kalman filter derivation but I can't see where such hypothesis is crucial.

Could you help me showing where the mathematical proof fails if I remove such hypothesis?

Thanks.

EDIT: I try to share my doubt more precisely.

Let's consider the formulas in Fig. 1 in this article. These two formulas have been obtained without assumptions on the process noise. Then, again without assumptions on the process noise, the author arrives at formula (5), in which appears the probability density $p(x_k | x_{k-1})$. Taking into account the first equation of the system (1), now he says that if we suppose the gaussianity and the whiteness of the process noise, we can then write such probability density as $\mathcal{N}(f(x_{k-1}),Q)$.

It is exactly in this point that my doubt arises: it seems to me that I could have written $$p(x_k | x_{k-1})=\mathcal{N}(f(x_{k-1}),Q)$$ even if I had only assumed gaussianity of the process noise, without whiteness, because $x_k$ is conditioned only by $x_{k-1}$ and not also by $q_{k-1},q_{k-2},...$.

Answer 1

In the section of the paper mentioned in the question (just below equation 5, page 3), the authors have mentioned that

By assuming that $w_k$ is a white Gaussian noise independent of $x_k$, we have $x_k | x_{k-1} \sim \mathcal{N}(f(x_{k-1}), Q)$

The above distribution is derived from the model described in equation 1, page 2, where it is stated that $x_{k+1} = f(x_k) + w_k$, along with other descriptions. To have the above distribution for a deterministic function $f(\cdot)$, three things are required: that $w_k$ is independent of $x_k$, that the distribution of $w_k$ is Gaussian with mean zero, and that the covariance matrix of $w_k$ does not vary over $k$ and remain fixed as $Q$. The whiteness of the Gaussian noise $w_k$ indicates the third condition, that the covariance matrix of $w_k$ is constant over $k$. This is somewhat different from the usual univariate white noise random variable. Here, the white noise is a multivariate random vector.