Random moment problems under constraints

We investigate moment sequences of probability measures on subsets of the real line under constraints of certain moments being fixed. This corresponds to studying sections of nth moment spaces, that is, the spaces of moment sequences of order n. By equipping these sections with the uniform or more general probability distributions, we manage to give for large n precise results on the (probabilistic) barycenters of moment space sections and the fluctuations of random moments around these barycenters. The measures associated to the barycenters belong to the Bernstein.Szeg. class and show strong universal behavior. We prove Gaussian fluctuations and moderate and large deviations principles. Furthermore, we demonstrate how fixing moments by a constraint leads to breaking the connection between random moments and random matrices.