INMA(1)型随机矩阵的极限谱分布

Limiting Spectral Distribution of INMA(1) Random Matrix

摘要: 整值时间序列数据在多个领域中广泛存在，如金融学、无线通讯网络、犯罪学等。对整值时间序列模型的研究在理论和应用上都有着重要价值。然而，面对当前海量的高维数据，经典的极限理论不再适用，亟需发展分析大维整值时间序列模型极限性质的理论与方法。从文章结构和研究意义整体上来看，本文研究了一类具有整值时间序列结构的大维随机矩阵的极限谱分布，即INMA(1)型随机矩阵的极限谱分布。先介绍了INMA(1)型随机矩阵的定义，并证明INMA(1)型随机矩阵样本协方差矩阵的极限谱分布的存在性；其次，针对文章中的内容和结果，利用Stieltjes变换给出了该模型的样本协方差矩阵的极限谱密度；最后通过数值模拟，验证了本文方法的有效性。

Abstract: Integer time series data are widely used in many fields; such as finance, wireless communication networks, criminology, etc. The study of integer-valued time series models has important value in both theory and application. However, in the face of the current massive high-dimensional data, classical bounds theory is no longer applicable, and there is an urgent need to develop theories and methods for analysing the bounds properties of high-dimensional integer time series models. From the perspective of the overall structure and research significance of the article, this paper investi-gates the limit spectral distribution of a class of high-dimensional random matrices with inte-ger-valued time-series structure, namely the limit spectral distribution of INMA(1) type random matrices. First, the definition of INMA(1) type random matrix is introduced, and the existence of the limit spectral distribution of the covariance matrix of the INMA(1) type random matrix sample is proved; Second, based on the contents and results of the article, the Stieltjes transform was used to obtain the maximum spectral density of the sample covariance matrix of the model; Finally, the ef-fectiveness of the proposed method was verified by numerical simulation.