论文标题
高斯观测的大型样品协方差矩阵，具有均匀相关性衰变
Large Sample Covariance Matrices of Gaussian Observations with Uniform Correlation Decay
论文作者
论文摘要
我们得出了$ v_n = \ frac {1} {n} xx^t $的样本协方差矩阵的Marchenko-Pastur（MP）定律，其中$ x $是a $ p \ times n $ data n $ data n $ data n \ $ p/n \ to y \ in（0，\ infty）$ n，\ infty $ n，as p n，p \ f \ f。我们假设$ x $中的数据来自相关的关节正态分布。特别是，相关性既可以跨行和$ x $的列来起作用，我们不假定特定的相关结构，例如可分离的依赖关系。取而代之的是，我们假设相关性以$ a_n/n $的速度均匀地收敛至零，在$ a_n $中，$ a_n $可能会略微生长到无穷大。我们采用瞬间的方法：我们确定了$ a_n $增长的确切条件，这将确保经验频谱分布（ESD）的矩汇合到MP时刻。如果不满足条件，我们可以构建一个合奏，几乎有限的ESD瞬间都有差异。我们还调查了$ c/n^δ$的均匀相关性结合下的$ v_n $的运算符规范，其中$ c，δ> 0 $是固定的，并在$δ= 1 $的情况下观察到相变。特别是，只有在$Δ> 1 $的情况下，只能保证操作员规范授予MP分布支持的最大支持。该分析导致了MP法律几乎肯定地拥有的示例，但是操作员规范在极限中仍然是随机的，我们提供了其确切的限制分布。
We derive the Marchenko-Pastur (MP) law for sample covariance matrices of the form $V_n=\frac{1}{n}XX^T$, where $X$ is a $p\times n$ data matrix and $p/n\to y\in(0,\infty)$ as $n,p \to \infty$. We assume the data in $X$ stems from a correlated joint normal distribution. In particular, the correlation acts both across rows and across columns of $X$, and we do not assume a specific correlation structure, such as separable dependencies. Instead, we assume that correlations converge uniformly to zero at a speed of $a_n/n$, where $a_n$ may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of $a_n$ which will guarantee that the moments of the empirical spectral distributions (ESDs) converge to the MP moments. If the condition is not met, we can construct an ensemble for which all but finitely many moments of the ESDs diverge. We also investigate the operator norm of $V_n$ under a uniform correlation bound of $C/n^δ$, where $C,δ>0$ are fixed, and observe a phase transition at $δ=1$. In particular, convergence of the operator norm to the maximum of the support of the MP distribution can only be guaranteed if $δ>1$. The analysis leads to an example for which the MP law holds almost surely, but the operator norm remains stochastic in the limit, and we provide its exact limiting distribution.
