On singular values of data matrices with general independent columns

In this paper, we analyse singular values of a large $p\times n$ data matrix $\mathbf{X}_n= (\mathbf{x}_{n1},\ldots,\mathbf{x}_{nn})$ where the column $\mathbf{x}_{nj}$'s are independent $p$-dimensional vectors, possibly with different distributions. Such matrices common in high-dimensional statistics. Under key assumption that covariance $\mathbf{\Sigma}_{nj}=\text{Cov}(\mathbf{x}_{nj})$ can be asymptotically simultaneously diagonalizable, and appropriate convergence their spectra, establish limiting distribution for $\mathbf{X}_n$ when both dimension $p$ $n$ grow to infinity comparable magnitude. The model goes beyond includes many existing works on types sample matrices, including weighted matrix, Gram linear times series models. Furthermore, develop two applications our general approach. First, obtain existence uniqueness new spectral realized multi-dimensional diffusion process anisotropic time-varying co-volatility processes. Secondly, derive recent matrix-valued auto-regressive model. Finally, generalized finite mixture model, is obtained.