协方差矩阵的渐近无穷小自由性
Asymptotic infinitesimal freeness of covariance matrices
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中文总结 AI 辅助
本文推导协方差矩阵混合矩的1/n展开，给出矩和无穷小矩显式公式，并在四阶矩条件下证明独立协方差矩阵的渐近自由性与无穷小自由性，推广了Wishart系综的结果。
中文摘要 AI 辅助
我们考虑$n\times n$协方差矩阵$M=\frac{1}{n}XX^*$，其中$X=(x_{i,j})$是一个元素为独立复随机变量的矩阵，满足$\mathbb{E}(x_{i,j})=0$和$\mathbb{E}(|x_{i,j}|^2)=1$。我们推导了混合矩$\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_q)}))$的$\frac{1}{n}$展开式，形式为$a_0+a_1\frac{1}{n}+O(\frac{1}{n^2})$。这使我们能够找到多个允许重复的协方差矩阵的矩和无穷小矩的显式公式。作为我们公式的一个应用，我们在四阶矩条件下推导了独立协方差矩阵的渐近自由性和无穷小自由性。这推广了之前关于$ x_{i,j}$为复高斯的Wishart系综的结果。
英文摘要
We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_q)}))$, of the form $a_0+a_1\frac{1}{n}+O(\frac{1}{n^2})$. This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which $x_{i,j}$ is complex Gaussian.
发表机构
- Tecnológico de Monterrey, School of Engineering and Sciences（蒙特雷理工学院，工程与科学学院）
- Department of Applied Mathematics, Chung Yuan Christian University（中原大学应用数学系）
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