Moderate Deviations for Extreme Eigenvalues of Real-Valued Sample Covariance Matrices

Consider the sample covariance matrices of form $$W=n^{-1}C C^{\top }$$ W = n - 1 C C ⊤ , where C is a $$k\times n$$ k × n matrix with real-valued, independent and identically distributed (i.i.d.) mean zero entries. When the squares of the i.i.d. entries have finite exponential moments, the moderate deviations for the extreme eigenvalues of W are investigated as $$n\rightarrow \infty $$ n → ∞ and either k is fixed or $$k\rightarrow \infty $$ k → ∞ with some suitable growth conditions. The moderate deviation rate function reveals that the right (left) tail of $$\lambda _{\max }$$ λ max is more like Gaussian rather than the Tracy–Widom type distribution when k goes to infinity slowly.