Edge and Outlier Eigenvalue Fluctuations

Why Eigenvalue Fluctuation Scales Matter

A sample eigenvalue close to the upper edge of the noise spectrum and one separated above that edge may both be estimates computed from the same covariance matrix. Their random errors, however, are not generally of the same order or governed by the same limiting law. At the edge, the natural scale is \(n^{-2/3}\) and the limiting distribution is Tracy–Widom. A separated outlier fluctuates on the larger \(n^{-1/2}\) scale and is asymptotically Gaussian.

The distinction matters when assessing whether an observed eigenvalue is compatible with the noise spectrum, or when reporting uncertainty for a detected component. The scaling statements below concern fixed, distinct regimes. A spike approaching the phase-transition threshold requires a separate critical-regime description.

Model and Notation

Let \(Z\) be a \(p\times n\) matrix whose entries are independent \(N(0,1)\) random variables. Consider the sample covariance matrix

where \(\Sigma\) is the population covariance matrix. Write its ordered sample eigenvalues as \(\lambda_1\geq\lambda_2\geq\cdots\), and suppose \(p,n\to\infty\) with \(p/n\to\gamma\), where \(0<\gamma<\infty\). For finite-sample centering, use \(\gamma_n=p/n\). In the unspiked case \(\Sigma=I_p\), the upper edge of the limiting Marchenko–Pastur spectrum is \(\lambda_+(\gamma)=(1+\sqrt{\gamma})^2\).

For a rank-one example, take \(\Sigma=\operatorname{diag}(\alpha,1,\ldots,1)\), with \(\alpha>0\). Its critical spike value is \(\alpha_c=1+\sqrt{\gamma}\). A fixed spike below this threshold does not create a separated upper outlier; a fixed spike above it does. The critical value is the same phase-transition threshold considered in the BBP eigenvalue phase transition.

Non-Outlier Eigenvalues: Tracy–Widom Edge Fluctuations

In the real Gaussian model, under the null or with a fixed finite number of population spikes strictly below the threshold, the upper non-outlier eigenvalues fluctuate at the spectral edge. Define

The largest non-outlier eigenvalue has the asymptotic form

where \(W_1\) has the Tracy–Widom distribution of type 1. The same edge scaling governs any fixed number of the top non-outlier eigenvalues jointly, though their limiting values are dependent. The statement uses the aspect ratio \(\gamma_n\) in the centering: replacing it by its limit \(\gamma\) without controlling the resulting centering error can invalidate an \(n^{-2/3}\)-scale approximation.

The scale coefficient follows by taking the large-\(n,p\) limit of the real Wishart edge scale. Up to centering terms negligible at order \(n^{-2/3}\), a finite-size scale is

Since \(p/n=\gamma_n\), the first factor is asymptotic to \(n^{-1/2}(1+\sqrt{\gamma_n})\), and the quantity in parentheses is asymptotic to \(n^{-1/2}(1+\gamma_n^{-1/2})\). Thus

In the last equality, \((1+\sqrt{\gamma_n})(1+\gamma_n^{-1/2})^{1/3}=(1+\sqrt{\gamma_n})^{4/3}\gamma_n^{-1/6}\). This gives both the exponent and the coefficient in the edge scale.

Separated Outliers: Gaussian Fluctuations

For the rank-one covariance spike \(\alpha\), a fixed supercritical value satisfies \(\alpha>1+\sqrt{\gamma}\). The corresponding sample outlier converges to

For a simple spike in the real Gaussian model, with \(\alpha\) fixed strictly above threshold, the centered fluctuation theorem is

Here \(\lambda_{\mathrm{out}}\) denotes the sample eigenvalue associated with that spike. The limiting variance is positive precisely in the fixed supercritical regime: \(\alpha>1+\sqrt{\gamma}\) implies \((\alpha-1)^2>\gamma\). The result requires an isolated, simple spike and a fixed positive separation from the threshold; it is not uniform as the spike approaches criticality.

Comparing the Two Scales

The distinction between the fluctuation rates can be checked directly. If the Gaussian variance coefficient for a fixed separated outlier is denoted by \(v>0\), its standard deviation is approximately \(\sqrt{v}\,n^{-1/2}\). The edge scale is \(c(\gamma)n^{-2/3}\). Their ratio is

Thus, as \(n\) increases, the scale of a separated outlier’s Gaussian fluctuations is larger than the edge scale, even though both fluctuations shrink to zero. More importantly, the eigenvalues are centered at different locations: an edge eigenvalue is centered at \(\lambda_+\), whereas an outlier is centered at \(\rho(\alpha,\gamma)>\lambda_+\). Comparing either fluctuation law without first identifying the appropriate center is misleading.

Common Mistake and Practical Interpretation

A frequent error is to use “the largest eigenvalue” as if it identified the fluctuation law by itself. It does not. The relevant questions are whether the population contains a fixed supercritical spike, whether the sample eigenvalue is separated from the limiting bulk edge, and whether the spike remains a fixed distance from threshold as dimensions grow.

• Wrong reasoning: “The sample eigenvalue is the largest one, so it must have Gaussian fluctuations.” The largest eigenvalue in an unspiked or fixed subcritical model is at the edge and has Tracy–Widom fluctuations.
• Correction: First determine whether an outlier exists. Use the edge law for a non-outlier edge eigenvalue and the Gaussian law only for a fixed, separated, simple supercritical spike under the stated model assumptions.
• Another pitfall: Centering at the limiting edge when the finite-sample aspect ratio has not been accounted for. At an \(n^{-2/3}\) scale, even a small centering error can matter.

These distinctions guide the interpretation of high-dimensional PCA spectra. An eigenvalue’s distance from the estimated noise edge is informative, but uncertainty depends on its regime. Edge fluctuations support edge-based calibration; an established separated outlier calls for its own location and Gaussian-scale uncertainty. Near the transition, a fixed-regime approximation can be especially unreliable.