  We consider the moment space $\mathcal{M}^{p}_{2n+1}$ of moments up to the
order $2n + 1$ of $p_n\times p_n$ real matrix measures defined on the interval
$[0,1]$. The asymptotic properties of the Hankel determinant $\{\log\det
(M_{i+j}^{p_n})_{i,j=0,\ldots,\lfloor nt\rfloor}\}_{t\in [0,1]}$ of a uniformly
distributed vector $(M_1,\dots ,M_{2n+1})^t\sim\mathcal{U}(\mathcal{M}_{2n+1})$
are studied when the dimension $n$ of the moment space and the size of the
matrices $p_n$ converge to infinity. In particular weak convergence of an
appropriately centered and standardized version of this process is established.
Mod-Gaussian convergence is shown and several large and moderate deviation
principles are derived. Our results are based on some new relations between
determinants of subblocks of the Jacobi-beta-ensemble,which are of their own
interest and generalize Bartlett decomposition-type results for the
Jacobi-beta-ensemble from the literature.
