Multivariate analysis

Let $M$ be a real $p\times n$ matrix whose columns are independent samples from the multivariate Gaussian distribution $N_p(\mu,\Sigma)$, where $\Sigma$ is positive definite. Center each row by its sample mean and form the sample covariance matrix

$$\widehat M_{ij}=M_{ij}-\frac{1}{n}\sum_{s=1}^{n}M_{is},\qquad S=\frac{1}{n}\widehat M\widehat M^{T}.$$

Write $l_1\geq\cdots\geq l_p\geq0$ for the eigenvalues of $S$. In the spiked model, for a fixed integer $k<p$, the ordered eigenvalues of $\Sigma$ satisfy

$$\eta_1\geq\eta_2\geq\cdots\geq\eta_k>\eta_{k+1}=\cdots=\eta_p=1.$$

The question is to characterize the asymptotic statistics of the sample eigenvalues $l_i$ as $p,n\to\infty$ with $p/n\to\gamma\in(0,\infty)$, and to determine how the population spikes affect their limiting locations and distributions. The question does not prescribe a scaling regime for the spikes, so it encompasses different regimes rather than specifying a single distributional limit. Deift, Some open problems in random matrix theory and the theory of integrable systems

Background

Random matrix theory entered theoretical physics through Wigner’s work on neutron-scattering resonances in the 1950s, while its statistical roots go back to early twentieth-century statisticians. In principal component analysis, the eigenvalues of the sample covariance matrix measure variance along principal directions; a few eigenvalues accounting for a large share of the total variance can indicate dimension reduction. When the numbers of variables and samples grow at comparable rates, random matrix asymptotics provide a framework for understanding these eigenvalues. Deift, Some open problems in random matrix theory and the theory of integrable systems

For identity population covariance, the empirical eigenvalues accumulate in the Marchenko–Pastur bulk, and the largest eigenvalue has Tracy–Widom fluctuations after suitable centering and scaling. A spiked population changes finitely many eigenvalues of the population covariance while leaving the rest at the common bulk value. The leading sample eigenvalue exhibits a phase transition: sufficiently strong spikes separate from the upper edge, whereas the critical regime requires a different distributional analysis. Deift, Some open problems in random matrix theory and the theory of integrable systems Limits of spiked random matrices II

Known results

With the aspect-ratio convention $p/n\to\gamma$, the cited account gives an almost-sure limit for a supercritical largest spike: if $\eta_1>1+\sqrt{\gamma}$, then the leading sample eigenvalue separates from the upper Marchenko–Pastur edge and

$$l_1\longrightarrow\eta_1\left(1+\frac{\gamma}{\eta_1-1}\right)>(1+\sqrt{\gamma})^2.$$

This is a location result for the stated supercritical regime, not a fluctuation law covering all spike strengths. Deift, Some open problems in random matrix theory and the theory of integrable systems

Near the transition, Bloemendal and Virág report limiting distributions for the largest eigenvalues of rank-one spiked real Wishart matrices and their general-$\beta$ analogues, resolving the Baik–Ben Arous–Péché conjecture in that setting. Their later rank-$r$ result treats real, complex, and quaternion Wishart matrices. In its Wishart formulation, the matrix is uncentered: for fixed $r$, its covariance is $\Sigma_{n,p}=\widetilde\Sigma_{n,p}\oplus I_{p-r}>0$, and it has $n$ degrees of freedom. Put $m_{n,p}=(n^{-1/2}+p^{-1/2})^{-2/3}$. If

$$m_{n,p}\bigl(1-\sqrt{n/p}(\widetilde\Sigma_{n,p}-1)\bigr)\longrightarrow W,$$

then its nonzero eigenvalues $\lambda_j$ satisfy, jointly for every finite collection of indices,

$$\frac{m_{n,p}^{2}}{\sqrt{np}}\bigl(\lambda_j-(\sqrt n+\sqrt p)^2\bigr)\Rightarrow-\Lambda_{j-1},$$

where $\Lambda_0\leq\Lambda_1\leq\cdots$ are the eigenvalues of the limiting operator $\mathcal H_{\beta,W}$; the real case corresponds to $\beta=1$. This theorem is a separate uncentered-Wishart result, not a distributional theorem directly for the centered matrix $S$ above. Limits of spiked random matrices I Limits of spiked random matrices II