Large deviations for random matricial moment problems

We consider the moment space Mn corresponding to p × p complex matrix measures defined on K (K = [0, 1] or K = T). We endow this set with the uniform law. We are mainly interested in large deviations principles (LDP) when n→∞. First we fix an integer k and study the vector of the first k components of a random element ofMn . We obtain a LDP in the set of k-arrays of p× p matrices. Then we lift a random element of Mn into a random measure and prove a LDP at the level of random measures. We end with a LDP on Carthéodory 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.