Limiting laws for extreme eigenvalues of large-dimensional spiked Fisher matrices with a divergent number of spikes

TL;DR

This paper studies the asymptotic behavior of the largest eigenvalues of high-dimensional spiked Fisher matrices when the number of spikes diverges and the spikes are unbounded, extending classical results to more complex regimes.

Contribution

It derives limiting laws for extreme eigenvalues of large-dimensional spiked Fisher matrices with a diverging number of unbounded spikes, a setting not previously fully analyzed.

Findings

Established limiting distributions for extreme eigenvalues in high-dimensional regimes.

Extended classical spiked model results to cases with diverging number of spikes.

Provided theoretical foundations for understanding eigenvalue behavior in complex high-dimensional settings.

Abstract

Consider the $p\times p$ matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to $p$, when two samples of sizes $n$ and $T$ from the two populations are available, we construct its corresponding sample version. In the regime of high dimension where both $n$ and $T$ are proportional to $p$, we investigate the limiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrix when the number of spikes is divergent and these spikes are unbounded.