Asymptotics of ultra-high-dimensional generalized spiked sample covariance matrix

This paper investigates the asymptotics of eigenstructure of sample covariance matrix under the spiked covariance matrix model in ultra-high-dimensional settings, where the dimensionality can grow much faster than the sample size with $p\asymp n^{\alpha }$, $\alpha >1$. We establish the first-order convergence limits of eigenvalue locations and eigenvector projections of properly scaled sample covariance matrix. Our results are extensions of \cite{bloemendal16,ding21}.