A unified approach to fluctuations of spectral statistics of generalized patterned random matrices

Kiran Kumar A. S., Shambhu Nath Maurya, Koushik Saha

Published 2024-02-06Version 1

An $N \times N$ generalized patterned random matrix is a symmetric matrix defined as $A=(x_{L(i,j)}\mathbf{1}_{\Delta}(i,j))$, where $\{x_k; k \in \mathbb{Z}^d\}$ is known as the input sequence, $L:\{1,2,\ldots, N\}^2 \rightarrow \mathbb{Z}^d$ is known as the link function and $\Delta \subseteq \{1,2,\ldots , N\}^2$. In 2008, Bose and Sen showed that under some restrictions on the link function $L$ and for $\Delta = \{1,2,\ldots , N\}^2$, the corresponding patterned matrices always have a sub-Gaussian limiting spectral distribution. Let $N: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function and $A_{n}=\left(x_{L(i, j)} \mathbf{1}_{\Delta}(i,j) \right)_{i,j=1}^{N(n)}$ be a sequence of generalized patterned matrices. We consider the LES of $A_n/\sqrt{N(n)}$ for the test function $\phi(x)=x^p$, given by $$ \frac{1}{\sqrt{N(n)}} \sum_{i=1}^{N(n)} \lambda_i^p=\eta_p, \mbox{ say}, $$ where $p$ is a positive integer and $\lambda_i$'s are the eigenvalues of $A_n/\sqrt{N(n)}$. Under some restrictions on $L$ and some moment assumptions on input entries, we show that when $p$ is even, $(\eta_p - \mathbb{E} [\eta_p])$ converges in distribution either to a normal distribution or to the degenerate distribution at zero. We show that under further assumptions on $L$, the limit is always a normal distribution. For odd degree monomial test functions, we derive the limiting moments of $\eta_p$ and show that the LES may not converge to a Gaussian distribution. We also study the LES for independent Brownian motion entries. We show that under some assumption on link function, and that if the limiting moment sequence of the LES of the generalized patterned random matrix with independent standard Gaussian entries is M-determinate, then the LES for Brownian motion entries converge to a deterministic real-valued process.