Large Deviations for Random Matricial Moment Problems

Abstract

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 $n\to \infty$. 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 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.

34 pages

References in corpus (4)

• The Analytic Theory of Matrix Orthogonal Polynomials
• Derivation of an eigenvalue probability density function relating to the Poincare disk
• Canonical moments and random spectral measures
• Matrix measures, random moments and Gaussian ensembles