Large Deviations for Random Matricial Moment Problems

TL;DR

This paper establishes large deviations principles for random matrix measures and related functions, using canonical moments to analyze asymptotic behaviors in complex matrix moment problems.

Contribution

It introduces a novel approach with matricial canonical moments to derive large deviations principles for matrix measures and associated functions.

Findings

LDP for the first k components of random matrix measures

LDP at the level of random matrix measures

LDP for Carathéodory and Schur functions related to the measures

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 $n \rightarrow \infty$. 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 Carth\'eodory 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.