Optimal Berry–Esseen rate under two-eigenvalue spectral conditions

Determine whether the optimal Berry–Esseen rate of n^{-1/2} can be achieved for Hilbert-valued nondegenerate U-statistics under bounded fourth moments and a covariance operator whose two largest eigenvalues are bounded away from zero.

Background

The paper establishes an n^{-1/8} Gaussian approximation rate for the Hilbert-space norm of nondegenerate U-statistics under bounded fourth moments and a minimal spectral condition requiring only two positive eigenvalues bounded away from zero. Earlier results achieve the faster n^{-1/2} rate under stronger assumptions, such as infinitely many or at least nine strictly positive eigenvalues. The authors explicitly leave unresolved whether the n^{-1/2} rate remains attainable under their weaker two-eigenvalue assumptions.