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arXiv 1002.4442math.PR
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随机矩阵幂的奇异值的渐近分布

Asymptotic distribution of singular values of powers of random matrices

Nikita Alexeev, Friedrich Götze, Alexander Tikhomirov

备注：
16 pages, 5 figures
发表信息：
Lithuanian Mathematical Journal, Vol. 50, No. 2, 2010, pp. 121-132
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摘要：

设$x$为复随机变量，使得${\E {x}=0}$，${\E |x|^2=1}$，${\E |x|^{4} < \infty}$。令$x_{ij}$、$i,j \in \{1,2,...\}$是$x$的独立拷贝。设${\Xb=(N^{-1/2}x_{ij})}$、$1\leq i,j \leq N$为随机矩阵。对于$\Xb$的伴随矩阵，写作$\Xb^*$，考虑乘积$\Xb^m{\Xb^*}^m$与一些$m \in \{1,2,...\}$。矩阵$\Xb^m{\Xb^*}^m$是Hermitian半正定的。令$λ_1,λ_2,...,λ_N$是$\Xb^m{\Xb^*}^m$的特征值（或矩阵$\Xb^m$的平方奇异值）。本文给出了经验分布函数\[ {F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{λ_k \leq x\}}}, \]的渐近分布函数\[ G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} \]，其中$\mathbb{I} \{A\}$表示事件$A$的指示函数。$G^{(m)}$的矩满足\[ M^{(m)}_p=\int_{\mathbb{R}}{x^p dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}. \]在自由概率论中$M^{(m)}_p$被称为Fuss-Catalan数。利用$m=1$，我们的结果变成了Marchenko-Pastur 1967的一个著名结果.

英文摘要：

Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$, ${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random matrix. Writing $\Xb^*$ for the adjoint matrix of $\Xb$, consider the product $\Xb^m{\Xb^*}^m$ with some $m \in \{1,2,...\}$. The matrix $\Xb^m{\Xb^*}^m$ is Hermitian positive semi-definite. Let $λ_1,λ_2,...,λ_N$ be eigenvalues of $\Xb^m{\Xb^*}^m$ (or squared singular values of the matrix $\Xb^m$). In this paper we find the asymptotic distribution function \[ G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} \] of the empirical distribution function \[ {F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{λ_k \leq x\}}}, \] where $\mathbb{I} \{A\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy \[ M^{(m)}_p=\int_{\mathbb{R}}{x^p dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}. \] In Free Probability Theory $M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko--Pastur 1967.

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