论文标题
相关正常人口的相关矩阵：最大的特征值的波动，批量特征和股票市场的缩放率
Correlation matrix of equi-correlated normal population: fluctuation of the largest eigenvalue, scaling of the bulk eigenvalues, and stock market
论文作者
论文摘要
给定$ n $维尺寸$ t $的样本，并形成样本相关矩阵$ \ mathbf {c} $。假设$ n $和$ t $倾向于无限，$ t/n $会收敛到固定有限常数$ q> 0 $。如果人口是一个因素模型，那么$ \ mathbf {c} $的特征值分布几乎肯定会微弱地收敛于marčenko-pastur分布，因此索引为$ q $，比例参数是特定方差与$ i $ $ $ $ $的限制比率（i $ thable $（i \ to \ infty）$。对于具有等值的系数$ρ$的$ n $维二维正常人，这是一种单因素型号，对于最大的特征值$λ$ $ \ mathbf {c} $，我们证明$λ/n $ converges to equi-cormis-equi-cormention cormentes of equi-cormiportiation $ρ$ρ$几乎是纯粹的。 These results suggest an important role of an equi-correlated normal population and a factor model in (Laloux et al. Random matrix theory and financial correlations, Int. J. Theor. Appl. Finance, 2000): the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of Marčenko-Pastur distribution of index $T/N $ and scale parameter $1-λ/N$.此外，我们提供了等应相关正常种群的样品协方差矩阵的最大特征值的限制分布。我们讨论了相关系数（$ n $）的衰减率的相变。
Given an $N$-dimensional sample of size $T$ and form a sample correlation matrix $\mathbf{C}$. Suppose that $N$ and $T$ tend to infinity with $T/N $ converging to a fixed finite constant $Q>0$. If the population is a factor model, then the eigenvalue distribution of $\mathbf{C}$ almost surely converges weakly to Marčenko-Pastur distribution such that the index is $Q$ and the scale parameter is the limiting ratio of the specific variance to the $i$-th variable $(i\to\infty)$. For an $N$-dimensional normal population with equi-correlation coefficient $ρ$, which is a one-factor model, for the largest eigenvalue $λ$ of $\mathbf{C}$, we prove that $λ/N$ converges to the equi-correlation coefficient $ρ$ almost surely. These results suggest an important role of an equi-correlated normal population and a factor model in (Laloux et al. Random matrix theory and financial correlations, Int. J. Theor. Appl. Finance, 2000): the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of Marčenko-Pastur distribution of index $T/N $ and scale parameter $1-λ/N$. Moreover, we provide the limiting distribution of the largest eigenvalue of a sample covariance matrix of an equi-correlated normal population. We discuss the phase transition as to the decay rate of the equi-correlation coefficient in $N$.
