论文标题
限制来自重尾种群样本相关矩阵特征值的分布
Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations
论文作者
论文摘要
考虑一个$ p $ - 二维群体$ {\ mathbf x} \ in \ mathbb {r}^p $，在吸引稳定分布的范围内，带有索引$α\ in（0,2）$。由于$ {\ Mathbf x} $的方差是无限的，因此示例协方差矩阵$ {\ MathBf S} _n = n = n^{ - 1} \ sum_ {i = 1}^n {{\ \ m马理bf x}} X} _1，\ ldots，{\ Mathbf X} _n $来自人群的行为不佳，有趣的是使用示例相关矩阵$ {\ Mathbf r} _n = \ {\ operatateOrname {\ operatatorname {diag}（diag}（diag}） {\ Mathbf S} _n \ {\ operatorname {diag}（{\ Mathbf S} _n）_n）\}^{ - 1/2} $。本文找到了$ {\ Mathbf r} _n $的特征值的限制分布，当尺寸$ p $和样本大小$ n $生长到无穷大，因此$ p/n \至γ\ in（0，\ infty）$。限制分布的家族$ \ {h_ {α，γ} \} $是新的，取决于两个参数$α$和$γ$。 $ h_ {α，γ} $的力矩被充分识别为两个贡献的总和：经典Marčenko-Pastur定律的第一个，第二个是由于沉重的尾巴。此外，家庭$ \ {h_ {α，γ} \} $在边界处具有连续的扩展，$α= 2 $和$α= 0 $，导致了Marčenko-Pastur定律和修改后的泊松分布。 我们的证明使用了矩的方法，即[18]中开发的路径缩短算法以及一些新颖的图计数组合学。结果，$ h_ {α，γ} $的力矩是用组合对象（例如第二种的stirling number）表示的。还提供了有关这些限制分布$ H_ {α，γ} $的模拟研究，以与Marčenko-Pastur定律进行比较。
Consider a $p$-dimensional population ${\mathbf x} \in\mathbb{R}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $α\in (0,2)$. Since the variance of ${\mathbf x}$ is infinite, the sample covariance matrix ${\mathbf S}_n=n^{-1}\sum_{i=1}^n {{\mathbf x}_i}{\mathbf x}'_i$ based on a sample ${\mathbf x}_1,\ldots,{\mathbf x}_n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix ${\mathbf R}_n= \{\operatorname{diag}({\mathbf S}_n)\}^{-1/2}\, {\mathbf S}_n \{\operatorname{diag}({\mathbf S}_n)\}^{-1/2}$. This paper finds the limiting distributions of the eigenvalues of ${\mathbf R}_n$ when both the dimension $p$ and the sample size $n$ grow to infinity such that $p/n\to γ\in (0,\infty)$. The family of limiting distributions $\{H_{α,γ}\}$ is new and depends on the two parameters $α$ and $γ$. The moments of $H_{α,γ}$ are fully identified as sum of two contributions: the first from the classical Marčenko-Pastur law and a second due to heavy tails. Moreover, the family $\{H_{α,γ}\}$ has continuous extensions at the boundaries $α=2$ and $α=0$ leading to the Marčenko-Pastur law and a modified Poisson distribution, respectively. Our proofs use the method of moments, the path-shortening algorithm developed in [18] and some novel graph counting combinatorics. As a consequence, the moments of $H_{α,γ}$ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions $H_{α,γ}$ is also provided for comparison with the Marčenko-Pastur law.
