ON SINGULAR VALUES OF DATA MATRICES WITH GENERAL INDEPENDENT COLUMNS

• 成果类型: Article
• 署名作者: Mei, By Tianxing; Wang, Chen; Yao, Jianfeng
• 署名单位: University of Hong Kong; The Chinese University of Hong Kong, Shenzhen
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2263
• 发表日期: 2023
• 页码: 624-645
• 关键词: empirical distribution; Covariance matrices; eigenvalues

摘要

We analyze the singular values of a large p x n data matrix Xn = (xn1, . . . ,xnn), where the columns {xnj } are independent p-dimensional vec-tors, possibly with different distributions. Assuming that the covariance ma-trices Enj = Cov(xnj) of the column vectors can be asymptotically simulta-neously diagonalized, with appropriately converging spectra, we establish a limiting spectral distribution (LSD) for the singular values of Xn when both dimensions p and n grow to infinity in comparable magnitudes. Our ma-trix model goes beyond and includes many different types of sample covari-ance matrices in existing work, such as weighted sample covariance matri-ces, Gram matrices, and sample covariance matrices of a linear time series model. Furthermore, three applications of our general approach are devel-oped. First, we obtain the existence and uniqueness of the LSD for realized covariance matrices of a multi-dimensional diffusion process with anisotropic time-varying co-volatility. Second, we derive the LSD for singular values of data matrices from a recent matrix-valued auto-regressive model. Finally, we also obtain the LSD for singular values of data matrices from a generalized finite mixture model.