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On the asymptotic distribution of the singular values of powers of random matrices (1012.2743v1)

Published 13 Dec 2010 in math.PR

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 $\mathbf W:={n{-\frac m2}} \mathbf Xm$ 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 $$ \alpha_k(m):=\int_{\mathbb R}xk\,d\,G(x)=\frac {1}{mk+1}\binom{km+k}{k}. $$

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