Large deviations for random matricial moment problems

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