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One observation:
$$A = I+L,$$ where $L$ is a lower triangular matrix with $0$ in the diagonals. This matrix $L$ can be seen to satisfy $L^n=0$, and $L^j\ne 0,\ 1\le j\le n-1$. Thus, one can write $$A^...

Just an idea.
I think the best way to start is to expand on the structure of the matrices $D, C$. For example, $D$ can be readily seen to be expressible in the form $$\alpha M_1\otimes I_2=\alpha\...

Let us consider the special case when $\mu$ is the Lebesgue measure on $\mathbb{R}$. The derivation then should be extendable to the general case without much effort.
We need to find the distribution ...

You can try applying gradient ascent method and run it for a few iterations to get an initial guess for $\mathbf{t}$. As the function is strongly concave, the gradient ascent is sure to converge to ...

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