On the asymptotic distribution of the singular values of powers of random matrices

TL;DR

This paper investigates the asymptotic behavior of the singular values of powers of large random matrices with independent entries, showing convergence of their spectral distribution to a specific limit distribution under certain conditions.

Contribution

It establishes the limiting spectral distribution for powers of random matrices with independent entries, extending understanding of their asymptotic singular value distribution.

Findings

Expected spectral distribution converges to a distribution defined by specific moments.

The moments are given explicitly by a combinatorial formula involving binomial coefficients.

Results hold under a Lindeberg condition for the fourth moment.

Abstract

We consider powers of random matrices with independent entries. Let $X_{ij}, i,j\ge 1$, be independent complex random variables with $\E X_{ij}=0$ and $\E |X_{ij}|^2=1$ and let $\mathbf X$ denote an $n\times n$ matrix with $[\mathbf X]_{ij}=X_{ij}$, for $1\le i, j\le n$. Denote by $s_1^{(m)}\ge...\ge s_n^{(m)}$ the singular values of the random matrix $\mathbf W:={n^{-\frac m2}} \mathbf X^m$ and define the empirical distribution of the squared singular values by $$ \mathcal F_n^{(m)}(x)=\frac1n\sum_{k=1}^nI_{\{{s_k^{(m)}}^2\le x\}}, $$ where $I_{\{B\}}$ denotes the indicator of an event $B$. We prove that under a Lindeberg condition for the fourth moment that the expected spectral distribution $F_n^{(m)}(x)=\E \mathcal F_n^{(m)}(x)$ converges to the distribution function $G^{(m)}(x)$ defined by its moments $$ \alpha_k(m):=\int_{\mathbb R}x^k\,d\,G(x)=\frac {1}{mk+1}\binom{km+k}{k}. $$