Optimal eigen expansions and uniform bounds

Let $\bigl\{X_k\bigr\}_{k \in \mathbb{Z}} \in \mathbb{L}^2(\mathcal{T})$ be a stationary process with associated lag operators ${\boldsymbol{\cal C}}_h$. 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, 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, we show convergence to an extreme value distribution, giving rise to the construction of simultaneous confidence sets. We also discuss how the asymptotic expansions transfer to the long-run covariance operator ${\boldsymbol{\cal G}}$ in a general framework.