Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, and preferably sparse) matrices arise naturally?

I am especially interested in problems that can be mapped onto a setup in which for each event of a reasonably nice point process on $\mathbb{R}$ (the simplest two such processes would be a Poisson or discrete-time process) there is an associated pair $(j,k) \in \{1,\dots,d\}^2$. In this case time-windowed sums $N_{jk}(t)$ of the various pairs can be formed in an obvious way (although there may be plenty of subtlety or freedom in the windowing itself): these supply such a matrix time series.

Each such pair $(j,k)$ could be regarded as a transition from server $j$ to another (possibly identical) server $k$ in a closed queue with $d$ servers and infinitely many clients. It is not hard to see that in the setting of communication networks, this framework amounts to a very general form of traffic analysis. Such an application should not be considered for an answer: it's already been covered.

A slightly more restrictive but simpler example is where the pairs $(j,k)$ are inherited from a càdlàg random walk on the root lattice

$A_n :=\left \{x \in \mathbb{Z}^{n+1} : \sum_{j=1}^{n+1} x_j = 0\right \}$.

Examples of this sort would also be of considerable interest to me.

Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, and preferably sparse) matrices arise naturally?

I am especially interested in problems that can be mapped onto a setup in which for each event of a reasonably nice point process on $\mathbb{R}$ (the simplest two such processes would be a Poisson or discrete-time process) there is an associated pair $(j,k) \in \{1,\dots,d\}^2$. In this case time-windowed sums $N_{jk}(t)$ of the various pairs can be formed in an obvious way (although there may be plenty of subtlety or freedom in the windowing itself): these supply such a matrix time series.

Each such pair $(j,k)$ could be regarded as a transition from server $j$ to another (possibly identical) server $k$ in a closed queue with $d$ servers and infinitely many clients. It is not hard to see that in the setting of communication networks, this framework amounts to a very general form of traffic analysis. Such an application should not be considered for an answer: it's already been covered.

A slightly more restrictive but simpler example is where the pairs $(j,k)$ are inherited from a cadlag random walk on the root lattice

$A_{d-1} :=\left \{x \in \mathbb{Z}^d : \sum_{j=1}^d x_j = 0\right \}$.

Examples of this sort would also be of considerable interest to me.