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 Mar\v{c}enko-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 $\rho$, which is a one-factor model, for the largest eigenvalue $\lambda$ of $\mathbf{C}$, we prove that $\lambda/N$ converges to the equi-correlation coefficient $\rho$ 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 Mar\v{c}enko-Pastur distribution of index $T/N $ and scale parameter $1-\lambda/N$. 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 $N$.

翻译：给定大小为$T$的$N$维样本，构成样本相关矩阵$\mathbf{C}$。假设$N$和$T$趋于无穷，且$T/N$收敛于固定有限常数$Q>0$。若总体为因子模型，则$\mathbf{C}$的特征值分布几乎必然弱收敛于指数为$Q$、尺度参数为特定方差与第$i$个变量（$i\to\infty$）之比的极限比率的Marčenko-Pastur分布。对于等相关系数为$\rho$的$N$维正态总体（即单因子模型），我们证明$\mathbf{C}$的最大特征值$\lambda$满足$\lambda/N$几乎必然收敛于等相关系数$\rho$。这些结果表明等相关系数正态总体与因子模型在Laloux等（Random matrix theory and financial correlations, Int. J. Theor. Appl. Finance, 2000）中起重要作用：股票收益率样本相关矩阵的特征值直方图拟合于指数$T/N$、尺度参数$1-\lambda/N$的Marčenko-Pastur分布密度。此外，我们给出了等相关系数正态总体样本协方差矩阵最大特征值的极限分布，并讨论了等相关系数关于$N$的衰减速率相关的相变现象。