Unbounded largest eigenvalue of large sample covariance matrices: Asymptotics, fluctuations and applications

Abstract

Given a large sample covariance matrix SN=1nΓN1/2ZNZN⁎ΓN1/2, where ZN is a N×n matrix with i.i.d. centered entries, and ΓN is a N×N deterministic Hermitian positive semidefinite matrix, we study the location and fluctuations of λmax(SN), the largest eigenvalue of SN as N,n→∞ and Nn−1→r∈(0,∞) in the case where the empirical distribution μΓN of eigenvalues of ΓN is tight (in N) and λmax(ΓN) goes to +∞. These conditions are in particular met when μΓN weakly converges to a probability measure with unbounded support on R+.We prove that asymptotically λmax(SN)∼λmax(ΓN). Moreover when the ΓN's are block-diagonal, and the following spectral gap condition is assumed:limsupN→∞λ2(ΓN)λmax(ΓN)<1, where λ2(ΓN) is the second largest eigenvalue of ΓN, we prove Gaussian fluctuations for λmax(SN)/λmax(ΓN) at the scale n.In the particular case where ZN has i.i.d. Gaussian entries and ΓN is the N×N autocovariance matrix of a long memory Gaussian stationary process (Xt)t∈Z, the columns of ΓN1/2ZN can be considered as n i.i.d. samples of the random vector (X1,…,XN)⊤. We then prove that ΓN is similar to a diagonal matrix which satisfies all the required assumptions of our theorems, hence our results apply to this case.