A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile

TL;DR

This paper establishes a Central Limit Theorem for the fluctuations of the log-determinant of Gram matrices with a variance profile, revealing Gaussian behavior and moment-dependent scaling in high-dimensional random matrix models.

Contribution

It provides the first CLT for the log-determinant of Gram matrices with a general variance profile, including explicit parameters and moment effects.

Findings

Centered and scaled log-determinant converges to a Gaussian distribution.

Additional terms in the variance appear when the fourth moment differs from Gaussian.

Results are relevant for wireless communication models.

Abstract

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$ Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}} X_{ij}^{n} $$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(\sigma_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$ \log\det(Y_n Y_n^* + \rho I_N) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\rho > 0$ is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4$^\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.