Marchenko–Pastur Law

For a T×N matrix X with i.i.d. N(0,σ²/T) entries, the empirical spectral distribution of the Wishart matrix W = X^TX converges (N,T→∞, N/T→γ) to the Marchenko-Pastur law ρ(λ) = (1-γ⁻¹)⁺δ(λ) + √((λ-λ₋)(λ₊-λ))/(2πγσ²λ). Spikes above λ+ = σ²(1+√γ)² signal true structure (BBP transition: spike eigenvalue detaches when signal > σ√γ).