Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

Characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications. To this end, in this article we introduce a general framework of asymptotic theory of eigenvectors for large spiked random matrices with diverging spikes and heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise. 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 establish asymptotic expansions for the general linear combination and further show that it is asymptotically normal after some normalization, where the weight vector can be arbitrary. We also provide a more general asymptotic theory for the spiked eigenvectors using the bilinear form. 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. for this article are available online.

Automated summary

This article introduces a general framework for the asymptotic theory of eigenvectors in large random matrices and establishes the asymptotic properties for spiked eigenvectors and eigenvalues in the scenario of generalized Wigner matrix noise. Simulation studies validate the theoretical results.