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Limit theorems for sample eigenvalues in a generalized spiked population model
Zhidong Bai, Jian-Feng Yao
TL;DR
The paper examines how spikes affect sample extreme eigenvalues while leaving the sample covariance matrix’s limiting spectral distribution unaffected. It extends prior convergence results to a generalized setting and introduces tools for establishing almost sure convergence of spike-associated sample eigenvalues.
Problem
The paper investigates the effect of spike eigenvalues on sample extreme eigenvalues when the limiting spectral distribution is unaffected by the spikes.
Method
The paper extends prior results to a generalized scheme and introduces mathematical tools for proving convergence of sample eigenvalues associated with spikes.
Results
The sample covariance matrix’s limiting spectral distribution remains unaffected by the spikes, while extreme sample eigenvalues associated with spikes almost surely converge to specified limits.
Takeaways & Limitations
The generalized framework identifies spike effects on sample extreme eigenvalues without changing the limiting spectral distribution of the sample covariance matrix.
Takeaways & Limitations
The analysis assumes convergence of the empirical spectral distribution of T_p and complex-valued random variables satisfying stated conditions.
Abstract
from arXiv · showhide
In the spiked population model introduced by Johnstone (2001),the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work (Bai and Yao, 2008), we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a {\em generalized} spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. New mathematical tools are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes.
1. Introduction
The paper extends spiked covariance asymptotics from an identity base population to a generalized model with an arbitrary base covariance spectrum. It identifies how finite-rank generalized spikes affect selected sample eigenvalues while leaving the limiting spectral distribution unchanged.
- The null model has population covariance T_p=I_p, with sample eigenvalues filling the Marčenko–Pastur support [a_y,b_y].
- A finite number of population spikes forms a finite-rank perturbation of the null case.
- The limiting spectral distribution remains the Marčenko–Pastur law despite the spikes.
- Earlier work established almost-sure limits and central limit theorems for extreme sample eigenvalues generated by ordinary spikes.
- The generalized model separates a small set of well-separated generalized spikes from an arbitrary base population spectrum with limiting distribution H.
- This paper identifies the effects of generalized spikes on particular sample eigenvalues and extends prior results to this generalized scheme.
- The paper develops almost-sure convergence results and then establishes a central limit theorem for the associated sample eigenvalues.
2. Generalized spiked population model
The generalized spiked population model treats a few population eigenvalues as generalized spikes separated from a base spectrum. The base empirical spectral distributions converge to H, which remains the limiting spectral distribution of the population covariance because the perturbation has finite rank.
- The model includes nonnegative Hermitian population components and spike eigenvalues α_1>⋯>α_K with specified multiplicities.
- The assumptions include i.i.d. complex entries, p′/n→y>0, and convergence of the population ESD sequence.
- The spike definition requires a positive separation from the relevant support, with d(α_k,Γ_H)>δ for all k≤K.
3. Known results on the spectrum of large sample covariance matrices
The paper reviews the Marčenko–Pastur limit for general sample covariance matrices and the analytic maps connecting population and sample spectral distributions. It also uses spectral gaps to establish exact separation and convergence properties for sample eigenvalues.
- The Marčenko–Pastur family is indexed by the aspect ratio y and population limit H.
- The map ψ_{y,H} and the Stieltjes transform characterize the relationship between the support of H and the generated Marčenko–Pastur distribution.
- 3.2. Exact separation of sample eigenvalues: Intervals outside the support of the associated Marčenko–Pastur distributions contain no sample eigenvalues eventually, almost surely.
- 3.2. Exact separation of sample eigenvalues: Under the separation conditions, transformed intervals contain no population eigenvalues and the corresponding sample eigenvalue counts match exactly.
4. Almost sure convergence of sample eigenvalues from generalized spikes
The paper classifies generalized spikes by the behavior of ψ′ and proves almost sure limits for their associated sample eigenvalues. Distant spikes generate outliers at ψ(αk), while close spikes converge either to boundary images or bulk quantiles.
- Spike classification: A generalized spike is distant when ψ′(α)>0 and close when ψ′(α)≤0.This derivative-based distinction determines whether the spike creates a separated sample-eigenvalue limit or remains tied to the bulk.
- Dependence on the base population: In the generalized model, the limiting behavior depends on the population spectrum H, the aspect ratio y, and the function ψ determined by them.As y decreases, a close spike for one Marčenko–Pastur law can become distant for another, showing that spike classification is model-dependent.
- Distant spikes: Distant generalized spikes produce nk consecutive sample eigenvalues that converge almost surely to ψ(αk).The result applies to spikes with a suitable interval where ψ′ is positive and places the associated sample eigenvalues outside the limiting spectral support.
- Close spikes: For close spikes, if ψ′>0 on a sub-interval, the associated nk sample eigenvalues converge to ψ(w), where w is the nearest endpoint of that interval.The endpoint is selected relative to the spike location within the maximal interval under consideration.
- Close spikes: If ψ′≤0 throughout the relevant interval, the associated nk sample eigenvalues converge almost surely to a bulk quantile.This occurs when the support gap disappears, so the spike-generated eigenvalues remain inside the limiting distribution rather than forming separated outliers.
5. CLT for sample eigenvalues from distant generalized spikes
The section develops a central limit theorem for sample eigenvalues generated by distant generalized spikes, extending methods for Johnstone’s spiked population model. The limiting eigenvalue distributions are Gaussian-matrix-based, generally non-Gaussian and asymptotically dependent, with Gaussian single-eigenvalue limits only for simple spikes.
- The paper derives a CLT for the nk-dimensional vector of sample eigenvalues associated with each distant generalized spike.
- The analysis follows Bai and Yao’s approach for Johnstone’s spiked population model, using the random form Kn and transforms of the limiting spectral distribution G.
- The proof framework identifies the limit distribution of the random matrices Rn(λ), with different real- and complex-valued cases and explicitly known Gaussian covariances.
- For each distant generalized spike, the centered sample eigenvalues converge weakly to the eigenvalues of a corresponding Gaussian random matrix block.
- The limiting distributions of packed extreme sample eigenvalues are generally non-Gaussian and asymptotically dependent.
- A single sample extreme eigenvalue has a Gaussian limiting distribution if and only if its corresponding generalized spike eigenvalue is simple.
6. Lemmas
The lemmas establish auxiliary analytic and probabilistic controls for the random forms used in the generalized spiked population analysis. They include integral transforms, event-based norm bounds, and an almost-sure law-of-large-numbers statement.
- The section defines the transforms m1(λ), m2(λ), and m3(λ) for λ outside ΓG.
- A lemma provides a law of large numbers for useful statistics of An, extending an earlier result from Johnstone’s model to the generalized model.
- Lemma 6.2 is stated under Theorem 4.1 for all λ in [a, b].
- Lemma 6.3 states that Kn(λ) converges almost surely to a specified limit for all λ in [a, b].
- On the event M, where S22 has no eigenvalues in [a′, b′], the norm of An is bounded using nested intervals [a, b] ⊂ (a′, b′) ⊂ (c, d).
- A quadratic-form average involving I + An converges almost surely to its trace-normalized counterpart.