Hankel determinants of random moment sequences

For $t \in [0,1]$ let $\underline{H}_{2\lfloor nt \rfloor} = ( m_{i+j})_{i,j=0}^{\lfloor nt \rfloor}$ denote the Hankel matrix of order $2\lfloor nt \rfloor$ of a random vector $(m_1,\ldots ,m_{2n})$ on the moment space $\mathcal{M}_{2n}(I)$ of all moments (up to the order $2n$) of probability measures on the interval $I \subset \mathbb{R}$. In this paper we study the asymptotic properties of the stochastic process $\{ \log \det \underline{H}_{2\lfloor nt \rfloor} \}_{t\in [0,1]}$ as $n \to \infty$. In particular weak convergence and corresponding large deviation principles are derived after appropriate standardization.