Optimal eigen expansions and uniform bounds

Let {Xk }k∈Z ∈ L2(T ) be a stationary process with associated lag operators
Ch. Uniform asymptotic expansions of the corresponding empirical eigenvalues and
eigenfunctions are established under almost optimal conditions on the lag operators
in terms of the eigenvalues (spectral gap). In addition, the underlying dependence
assumptions are optimal in a certain sense, including both short and long memory
processes. This allows us to study the relative maximum deviation of the empirical
eigenvalues under very general conditions. Among other things, convergence to an
extreme value distribution is shown. We also discuss how the asymptotic expansions
transfer to the long-run covariance operator G in a general framework.