Abstract

Characterizing the exact asymptotic distributions of high-dimensional eigenvectors for large structured random matrices poses important challenges yet can provide useful insights into a range of applications. This paper establishes the asymptotic properties of the spiked eigenvectors and eigenvalues for the generalized Wigner random matrix, where the mean matrix is assumed to have a low-rank structure. Under some mild regularity conditions, we provide the asymptotic expansions for the spiked eigenvalues and show that they are asymptotically normal after some normalization. For the spiked eigenvectors, we provide novel asymptotic expansions for the general linear combination and further show that the linear combination is asymptotically normal after some normalization, where the weight vector can be an arbitrary unit vector. Simulation studies verify the validity of our new theoretical results. Our family of models encompasses many popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis, and our general theory can be exploited for statistical inference in these large-scale applications. This is a joint work with Jianqing Fan, Yingying Fan and Xiao Han.