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

N. Alexeev ; F. Götze ; A. Tikhomirov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 9-42

Résumé

We consider powers of random matrices with independent entries. Let $X_{ij}$, $i,j\ge1$, be independent complex random variables with $\mathbf EX_{ij}=0$ and $\mathbf E|X_{ij}|^2=1$ 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 $s_1^{(m)}\ge\ldots\ge s_n^{(m)}$ 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 that the expected spectral distribution $F_n^{(m)}(x)=\mathbf E\mathcal F_n^{(m)}(x)$ converges under a Lindeberg condition to the distribution function $G^{(m)}(x)$ defined by its moments $$ \alpha_k(m):=\int_\mathbb Rx^k\,dG(x)=\frac1{mk+1}\binom{km+k}k. $$