On singular values of data matrices with general independent columns

Abstract

We analyze the singular values of a large p × n data matrix Xn = (xn1,...,xnn), where the columns {xnj} are independent p-dimensional vectors, possibly with different distributions. Assuming that the covariance matrices Σnj = Cov(xnj) of the column vectors can be asymptotically simultaneously 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 matrix model goes beyond and includes many different types of sample covariance matrices in existing work, such as weighted sample covariance matrices, Gram matrices, and sample covariance matrices of a linear time series model. Furthermore, three applications of our general approach are developed. 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.