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

Source abstract

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 $φ t\} \to 0$ 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 $λ_-$.