Assume we have a linear state-space model: $$ z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + Bu + w_{k}, $$ where $u$ is some control variable (constant intercept is the simplest case).

Kalman filter, for example, provides us the estimate of $E[x_{n}|z_{0}, \dots, z_{n}]$.

Assume that the state process is weakly stationary.

The question: what can one say about provided by Kalman filter estimate $\hat{x}_{k|k}$ of $E[x_{n}|z_{0}, \dots, z_{t}]$, if $k\to\infty$?

The first wrong idea was that it should converge (in probability) to $E[x]$. This is wrong and can be check on simulations given in

Kalman filter for AR(1) plus noise

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    $\begingroup$ Add self-study tag? $\endgroup$ – Jarle Tufto Nov 27 '19 at 16:16
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    $\begingroup$ Before voting to close as off topic, why not clarify whether this actually is self-study or not? I also do not get another close vote regarding subject-matter off-topicness; this looks entirely on topic to me. $\endgroup$ – Richard Hardy Nov 28 '19 at 8:53

As $k\to\infty$, we have an asymptotic gain, $K$ and we end up with linear time-invariant filter: $$ \hat{x}_{k|k} = F \hat{x}_{k-1|k-1} + K[z_{k} - HF\hat{x}_{k-1|k-1}], $$ see, for example, in wiki: https://en.wikipedia.org/wiki/Kalman_filter#Underlying_dynamical_system_model. Conclusion: $\hat{x}_{k|k}$ converges in distribution to some normal random variable.

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