CLT for linear spectral statistics of random matrix $S^{-1}T$

TL;DR

This paper establishes a central limit theorem for linear spectral statistics of the inverse of a random matrix multiplied by a fixed Hermitian matrix, broadening understanding of spectral behavior in complex random matrix models.

Contribution

It introduces a CLT for linear spectral statistics of the matrix $S^{-1}T$ with a general non-negative definite, non-random Hermitian matrix $T$, extending existing theoretical frameworks.

Findings

Proves a CLT for spectral statistics of $S^{-1}T$.

Applicable to a wide class of non-negative definite matrices.

Enhances theoretical understanding of spectral distributions in random matrix theory.

Abstract

This paper proposes a CLT for linear spectral statistics of random matrix $S^{-1}T$ for a general non-negative definite and {\bf non-random} Hermitian matrix $T$.