  We consider the moment space $\mathcal{M}_n^{K}$ corresponding to $p \times
p$ complex matrix measures defined on $K$ ($K=[0,1]$ or $K=\D$). We endow this
set with the uniform law. We are mainly interested in large deviations
principles (LDP) when $n \rightarrow \infty$. First we fix an integer $k$ and
study the vector of the first $k$ components of a random element of
$\mathcal{M}_n^{K}$. We obtain a LDP in the set of $k$-arrays of $p\times p$
matrices. Then we lift a random element of $\mathcal{M}_n^{K}$ into a random
measure and prove a LDP at the level of random measures. We end with a LDP on
Carth\'eodory and Schur random functions. These last functions are well
connected to the above random measure. In all these problems, we take advantage
of the so-called canonical moments technique by introducing new (matricial)
random variables that are independent and have explicit distributions.
