论文标题
样品协方差矩阵的特征值统计的中央限制定理与随机种群
Central limit theorem for eigenvalue statistics of sample covariance matrix with random population
论文作者
论文摘要
考虑示例协方差矩阵$$σ^{1/2} xx^tς^{1/2} $$ 其中$ x $是带有独立条目的$ m \ times n $随机矩阵，$σ$是$ m \ times m $对角矩阵。众所周知，如果$σ$是确定性的，那么 $$ \ sum_if（λ_i）$$ 分布收敛到高斯分布。这里$ \ {λ_i\} $是$σ^{1/2} xx^tς^{1/2} $和$ f $的eigenvalues，而$ f $是一个足够好的测试功能。在本文中，我们认为$σ$是随机的，并表明了 $$ \ frac {1} {\ sqrt n} \ sum_if（λ_i）$$ 分布收敛到高斯分布。这种现象意味着$σ$的随机性降低了$ \ {λ_i\} $之间的相关性。
Consider the sample covariance matrix $$Σ^{1/2}XX^TΣ^{1/2}$$ where $X$ is an $M\times N$ random matrix with independent entries and $Σ$ is an $M\times M$ diagonal matrix. It is known that if $Σ$ is deterministic, then the fluctuation of $$\sum_if(λ_i)$$ converges in distribution to a Gaussian distribution. Here $\{λ_i\}$ are eigenvalues of $Σ^{1/2}XX^TΣ^{1/2}$ and $f$ is a good enough test function. In this paper we consider the case that $Σ$ is random and show that the fluctuation of $$\frac{1}{\sqrt N}\sum_if(λ_i)$$ converges in distribution to a Gaussian distribution. This phenomenon implies that the randomness of $Σ$ decreases the correlation among $\{λ_i\}$.
