Marchenko-Pastur Law and Random Matrix Eigenvalue Density

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Random Matrix Theory / High-Dimensional Stats.

Your Task
Let $X \in \mathbb{R}^{n \times p}$ be a matrix of i.i.d. random variables with mean 0 and variance $\sigma^2 = 1$. Let $S = \frac{1}{n} X^T X$ be the sample covariance matrix. As $p, n \to \infty$ with aspect ratio $q = p / n \in (0, 1]$:
(1) State the Marchenko-Pastur probability density function for the empirical eigenvalue distribution of $S$.
(2) Identify the theoretical upper and lower spectral bounds $\lambda_{\pm} = (1 \pm \sqrt{q})^2$.
(3) Explain how quantitative hedge funds use this law to separate signal from noise in asset correlation matrices.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.