arXiv subjects

Haozhu Zhao

Publications and source records attributed to Haozhu Zhao.

2 recordsLinked to original sources

arXiv2026

Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices

We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a $p_n \times n$ data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime $p_n / n \to ϕ\in (0, \infty)$, the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is $k_* (ϕ) = \lfloor \sqrtϕ \rfloor + 2$, yielding the critical tail exponent $α_* (ϕ) = 2 + 2 / k_* (ϕ)$. This exponent decreases in steps as $ϕ$ increases, creating a staircase boundary between convergence to the upper Marčenko--Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.

math.PR↗

arXiv2026

Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices

We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $ξ$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to ϕ\in (0, \infty) \setminus \{1\}$. In the tall regime $ϕ> 1$, we prove almost-sure convergence of $λ_n (\mathbf{R}_n)$ to the lower Marčenko--Pastur edge $λ_- = (1 - \sqrtϕ)^2$ without additional moment assumptions. In the wide regime $ϕ< 1$, we establish a phase transition at the third-order tail scale. If $t^3 \mathbb{P}\{\lvert ξ\rvert > t\} \to 0$ as $t \to \infty$, the smallest eigenvalue $λ_{p_n} (\mathbf{R}_n)$ converges in probability to $λ_-$. While if $t^3 \mathbb{P}\{\lvert ξ\rvert > t\} \to \infty$, then $λ_{p_n} (\mathbf{R}_n)$ converges in probability to zero. At the critical scale $t^3 \mathbb{P}\{\lvert ξ\rvert > t\} \to κ\in (0, \infty)$, the point process of eigenvalues in the lower gap $(0, λ_-)$ converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of $λ_{p_n} (\mathbf{R}_n)$, which has a continuous density on $(0, λ_-)$ and a positive atom at $λ_-$.

math.PR↗