论文标题
关于大维滞后样品自相关矩阵的奇异值
On singular values of large dimensional lag-tau sample autocorrelation matrices
论文作者
论文摘要
我们研究了高维因子模型中lag-$τ$样本自动相关矩阵$ \ bf {r}_τ^ε$在高维因子模型中的界限自动相关矩阵$ \ bf {r}_τ^ε$。我们建立了限制光谱分布（LSD），该光谱分布（LSD）表征了$ \ bf {r}_τ^ε$的全局光谱，并得出了其最大的单数值的极限。所有渐近结果均在高维渐近状态下得出，其中数据维度和样本量按比例地归为无穷大。在温和的假设下，我们表明$ \ bf {r}_τ^ε$的LSD与lag-$τ$样本自动稳定矩阵的LSD相同。基于这种渐近等效性，我们还表明，$ \ bf {r}_τ^ε$的最大奇异值几乎可以肯定地收敛到其LSD支持的右端。我们的结果迈出了第一步，使用滞后$τ$样品自动相关矩阵确定因子分析中的因素数量。我们的理论结果也得到了数值实验的充分支持。
We study the limiting behavior of singular values of a lag-$τ$ sample auto-correlation matrix $\bf{R}_τ^ε$ of error term $ε$ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) which characterizes the global spectrum of $\bf{R}_τ^ε$, and derive the limit of its largest singular value. All the asymptotic results are derived under the high-dimensional asymptotic regime where the data dimension and sample size go to infinity proportionally. Under mild assumptions, we show that the LSD of $\bf{R}_τ^ε$ is the same as that of the lag-$τ$ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of $\bf{R}_τ^ε$ converges almost surely to the right end point of the support of its LSD. Our results take the first step to identify the number of factors in factor analysis using lag-$τ$ sample auto-correlation matrices. Our theoretical results are fully supported by numerical experiments as well.
