Hankel determinants of random moment sequences

TL;DR

This paper investigates the asymptotic behavior of the log-determinants of Hankel matrices formed from random moment sequences, establishing weak convergence and large deviation principles as the matrix size grows.

Contribution

It provides the first analysis of the asymptotic distribution and deviation principles for Hankel determinants of random moment sequences.

Findings

Weak convergence of the process of log-determinants.

Large deviation principles established.

Asymptotic normality under standardization.

Abstract

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.