论文标题
高维样品相关矩阵的最大和最小特征值的几乎确定收敛
Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices
论文作者
论文摘要
在本文中，我们表明，样本相关矩阵的最大和最小的特征值来自$ n $独立观察，与IID组件的$ p $二维时间序列肯定会肯定地汇聚到$（1+ \sqrtγ）^2 $和$（1- \sqrtγ）和$（1- \sqrtγ）$ n $ n. γ\在（0,1] $中，入口分布的截断差异是“几乎缓慢变化的”，我们通过自相应总和的力矩属性描述的条件。此外，这些样品相关矩阵的经验频谱分布薄弱地收敛，概率1，概率1，范围为Marchenko-Pastur Law，将其扩展到BAIE和ZH的行为（2008）。样本协方差和样品相关矩阵的特征值，并认为后者似乎更健壮，尤其是在无限的第四刻中，我们简要解决了一些实用问题，以估计一项模拟研究中的极端特征值。 在我们的证据中，我们使用矩与路径缩短算法的方法，该算法有效地使用样品相关矩阵的结构来计算矩阵规范的精确边界。我们认为，这种新方法可以在随机矩阵理论中进一步使用。
In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrtγ)^2$ and $(1-\sqrtγ)^2$, respectively, as $n \to \infty$, if $p/n\to γ\in (0,1]$ and the truncated variance of the entry distribution is 'almost slowly varying', a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marchenko-Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study. In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.
