Canonical Moments and Random Spectral Measures

We study some connections between the random moment problem and random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A, e), where A is a random matrix from a classical ensemble, and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix increases. The rate function for these large deviations involves the reversed Kullback information.