Large deviations for random matricial moment problems

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