How do heavy-tailed sample correlation matrices produce staircase eigenvalue transitions?

What the staircase boundary separates

The paper studies the largest eigenvalue λ₁(R_n) of the uncentered sample correlation matrix R_n = D_n^{-1/2} S_n D_n^{-1/2}, built from a p_n × n data matrix X_n with i.i.d. real entries of mean zero and unit variance, in the proportional regime p_n/n → φ ∈ (0, ∞). The empirical spectral distribution of R_n converges to the Marčenko–Pastur law with upper edge λ+ = (1+√φ)², but that alone does not fix the extreme eigenvalue. The paper's question is which tail conditions on the entry distribution ξ make λ₁(R_n) converge to λ+ and which make it separate as an outlier above λ+.

The answer is organized by a single integer, the first collision order k*(φ) = ⌊√φ⌋+2, and the associated critical tail exponent α*(φ) = 2 + 2/k*(φ). If t^{α*(φ)} P{|ξ|>t} → 0 as t → ∞, λ₁(R_n) converges in probability to λ+ = (1+√φ)². If instead the tail is supercritical at some order k ≥ k*(φ) and subcritical at the next order k+1, λ₁(R_n) converges in probability to the deterministic level θ_φ(k) = k(1 + φ/(k−1)). The boundary between these behaviors is not a single curve but a staircase, because k*(φ) increases at φ = 1, 4, 9, … and α*(φ) correspondingly decreases in steps.

Why collisions in one column drive the transition

The mechanism is a collision of large entries in distinct rows of a common column. The paper counts sets of k entries lying in k distinct rows of the same column, all exceeding √n in absolute value. The expected number of such k-row collisions is n · C(p_n, k) · P{|ξ|>√n}^k, which behaves like (φ^k/k! + o(1)) · (n^{α_k/2} P{|ξ|>√n})^k with α_k = 2 + 2/k. This identity is what makes the exponent α_k = 2 + 2/k the natural scale for order-k collisions.

The tail of ξ is then called subcritical, supercritical, or critical at order k according to whether t^{α_k} P{|ξ|>t} tends to zero, infinity, or a finite positive constant as t → ∞. The first collision order k*(φ) is the smallest integer that exceeds the threshold 1+√φ, which is why k*(φ) = ⌊√φ⌋+2 and why the critical exponent α*(φ) = 2 + 2/k*(φ) changes at φ = 1, 4, 9, …. In this picture, the number k of colliding rows plays the role that a population spike plays in spiked population models, and the outlier map θ_φ(x) = x(1 + φ/(x−1)) for x ≥ 1+√φ translates that collision order into the limiting location of the outlier.

The first few steps of the staircase

The staircase is easiest to read from the first few values. For 0 < φ < 1, k*(φ) = 2 and α*(φ) = 3. For 1 ≤ φ < 4, k*(φ) = 3 and α*(φ) = 8/3. For 4 ≤ φ < 9, k*(φ) = 4 and α*(φ) = 5/2. Each time φ crosses a perfect square, the first collision order rises by one and the critical exponent falls, so the subcritical region in the (φ, α) plane is bounded by a descending staircase rather than a smooth curve.

The paper also notes that the subcritical condition t^{α*(φ)} P{|ξ|>t} → 0 is strictly weaker than the moment assumption E|ξ|^{α*(φ)} < ∞. A tail comparable to t^{−α*(φ)}/log t satisfies the condition even though E|ξ|^{α*(φ)} = ∞. This matters for readers who would otherwise treat a finite moment of order α*(φ) as the relevant threshold; the paper's criterion is a tail-limit statement, not a moment-existence statement.

Deterministic outlier levels between the steps

Between the steps, the largest eigenvalue does not stay at λ+. If the tail is supercritical at order k ≥ k*(φ) and subcritical at order k+1, meaning t^{α_k} P{|ξ|>t} → ∞ and t^{α_{k+1}} P{|ξ|>t} → 0, then λ₁(R_n) converges in probability to θ_φ(k) = k(1 + φ/(k−1)). Two concrete consequences are stated: for 0 < φ < 1 with t³ P{|ξ|>t} → ∞ and t^{8/3} P{|ξ|>t} → 0, λ₁(R_n) → 2(1+φ); for 1 ≤ φ < 4 with t^{8/3} P{|ξ|>t} → ∞ and t^{5/2} P{|ξ|>t} → 0, λ₁(R_n) → 3(1+φ/2).

These levels are deterministic, not random, and they sit strictly above λ+ in the interior of the corresponding band. On the right vertical boundary φ = (k−1)², the level θ_φ(k) coincides with λ+, and because k*(φ) = k+1 on that boundary, the subcritical theorem applies there for α_{k+1} < α ≤ α_k. The paper is explicit that each right vertical boundary belongs to the subcritical regime, including the upper endpoint marked by an open circle but excluding the lower endpoint marked by a filled circle; the lower endpoint at α = α_{k+1} lies in the critical regime of order k+1.

Critical scales produce Poisson outliers

At the exact critical scale, where t^{α_k} P{|ξ|>t} → ω ∈ (0, ∞) for a fixed k ≥ k*(φ), the behavior changes again. The paper defines a spectral baseline β_φ(k−1) = (1+√φ) ∨ (k−1) and ϑ_φ(k−1) = θ_φ(β_φ(k−1)). At the first critical order k = k*(φ), this baseline equals the upper MP edge λ+; at higher orders it equals θ_φ(k−1). The point process of eigenvalues of R_n above ϑ_φ(k−1) then converges vaguely on (ϑ_φ(k−1), ∞) to a Poisson random measure with an atomless intensity Λ_{k,φ,ω} of finite positive total mass.

The intensity is built from k-row collisions with normalized entry magnitudes a ∈ R_+^k. For each such collision the paper forms the matrix M_k(a) with unit diagonal and largest eigenvalue γ_k(a), and the resulting outlier has asymptotic location θ_φ(γ_k(a)) whenever γ_k(a) > β_φ(k−1). The limiting distribution H_{k,φ,ω} of λ₁(R_n) in this critical regime is 0 below ϑ_φ(k−1), equals exp(−Λ_{k,φ,ω}((λ, ∞))) on [ϑ_φ(k−1), θ_φ(k)), and equals 1 at and above θ_φ(k). Because the intensity has finite positive total mass, H_{k,φ,ω} has a positive atom at the baseline ϑ_φ(k−1), which is what connects adjacent phases.

When finite variance is not enough

The paper's third regime is the ultra-supercritical one: if t^{α_k} P{|ξ|>t} → ∞ for every fixed k ≥ k*(φ), then λ₁(R_n) diverges in probability to infinity. The paper emphasizes that this condition can hold even when ξ has finite variance. A centered distribution with P{|ξ|>t} ≍ t^{−2}(log t)^{−2} has a finite second moment and satisfies the ultra-supercritical condition, so finite variance alone does not keep λ₁(R_n) bounded.

This is the sharpest contrast in the paper with the classical Bai–Yin picture for sample covariance matrices, where E|ξ|⁴ < ∞ gives almost sure convergence of λ₁(S_n) to λ+ and E|ξ|⁴ = ∞ gives lim sup λ₁(S_n) = ∞ almost surely. For the self-normalized correlation matrix R_n, the relevant threshold is not a fourth moment but the staircase of tail exponents α*(φ), α_{k*(φ)+1}, … determined jointly by the entry tail and the aspect ratio φ.

Where the staircase applies and where it stops

The results are stated for the uncentered correlation matrix R_n under Eξ = 0, E|ξ|² = 1, and p_n/n → φ ∈ (0, ∞), and the paper extends them to the Pearson sample correlation matrix R°_n, which centers each feature by its sample mean before normalization. The Pearson versions of Theorems 1.1, 1.2, 1.5, and 1.6 hold with the same critical intensity and limiting law. The paper also notes that the correlation matrix is invariant under positive rescaling of individual rows, so the unit-variance normalization entails no loss of generality for zero-mean, finite-variance entries.

The paper draws an analogy with a staircase phenomenon in the spectral deviations of random regular graphs, where the polynomial decay exponent of P{‖A_n‖ ≥ ρ(m)+ε} increases in integer steps as the spectral threshold rises. The paper is careful to say that the two staircases describe different phenomena: its own results concern changes in the first-order behavior of the largest eigenvalue as the entry tail and aspect ratio vary, whereas the random-regular-graph result concerns rare-event probabilities as the spectral threshold varies. The analogy lies in the local structures that produce distinct outlier levels, not in the quantities being compared.

For a reader deciding what to take from the paper, the operative object is the pair (φ, α). The first collision order k*(φ) = ⌊√φ⌋+2 fixes which tail exponent α*(φ) = 2 + 2/k*(φ) separates convergence to λ+ from an outlier, and the deterministic levels θ_φ(k) = k(1 + φ/(k−1)) tell you where the outlier sits once the tail is supercritical at order k and subcritical at k+1. The most consequential single check is whether t^{α*(φ)} P{|ξ|>t} tends to zero, infinity, or a finite positive constant, because that choice selects the subcritical, ultra-supercritical, or critical regime and therefore whether λ₁(R_n) stays at the MP edge, diverges, or follows a Poisson-driven limiting law with an atom at the baseline.

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