论文标题
大型样品相关矩阵的限制光谱分布
Limiting spectral distribution for large sample correlation matrices
论文作者
论文摘要
在本文中，我们考虑了样品相关矩阵的经验光谱分布，并在尺寸和样本量以相同速率增加时研究了对数据分布的轻度假设下的渐近行为。首先，我们为限制频谱分布提供了一个表征，以遵循Marchenko-Pastur定律，假设基础数据矩阵由I.I.D.组成。条目。随后，当允许在数据矩阵列内建立依赖性结构时，我们提供样品相关矩阵的极限光谱分布。与以前的工作相反，数据的第四刻可能是无限的，从而产生了根本的结构差异。更准确地说，引入了通过样品协方差伴侣分解近似样品相关矩阵的标准论点，并引入了解决样品相关矩阵的具有挑战性依赖关系结构的新技术。
In this paper, we consider the empirical spectral distribution of the sample correlation matrix and investigate its asymptotic behavior under mild assumptions on the data's distribution, when dimension and sample size increase at the same rate. First, we give a characterization for the limiting spectral distribution to follow a Marchenko-Pastur law assuming that the underlying data matrix consists of i.i.d. entries. Subsequently, we provide the limiting spectral distribution of the sample correlation matrix when allowing for a dependence structure within the columns of the data matrix. In contrast to previous works, the fourth moment of the data may be infinite, resulting in a fundamental structural difference. More precisely, the standard argument of approximating the sample correlation matrix by its sample covariance companion breaks down and novel techniques for tackling the challenging dependency structure of the sample correlation matrix are introduced.
