Asymptotic Theory of Eigenvectors for Random Matrices with Diverging Spikes

Area of Science:

• Mathematics
• Statistics
• Data Science

Background:

• Analyzing large random matrices is crucial for statistical applications.
• Understanding eigenvector distributions in these matrices presents significant challenges.
• Existing theories often lack generality for complex matrix structures.

Purpose of the Study:

• To develop a general framework for the asymptotic theory of eigenvectors (ATE) for large spiked random matrices.
• To establish asymptotic properties of spiked eigenvectors and eigenvalues under generalized Wigner matrix noise.
• To provide a robust theoretical foundation for statistical inference in large-scale data analysis.

Main Methods:

• Introduction of a novel general framework for asymptotic theory of eigenvectors (ATE).
• Establishment of asymptotic properties for spiked eigenvectors and eigenvalues with diverging spikes and heterogeneous variances.
• Derivation of asymptotic expansions and normality for spiked eigenvalues and eigenvectors.

Main Results:

• Asymptotic normality is established for spiked eigenvalues after normalization.
• Asymptotic expansions are derived for general linear combinations of spiked eigenvectors.
• The theory is validated through simulation studies, demonstrating its practical applicability.

Conclusions:

• The developed ATE framework provides a powerful tool for analyzing large spiked random matrices.
• The findings are applicable to diverse models like stochastic block models and topic models.
• This research facilitates advanced statistical inference in network and text analysis.