Empirical eigen expansions and uniform bounds

Let $\bigl\{X_k\bigr\}_{k \in \mathbb{Z}} \in \mathbb{L}^2(T)$ be a stationary process with associated covariance operator ${\boldsymbol{\cal C}}$. Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under optimal dependence assumptions, including both short and long memory processes. In addition, the underlying conditions on the covariance operator (spectral gap) are almost optimal. 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. Uniform rates of convergence for the relative trace and inverse trace of ${\boldsymbol{\cal C}}$ are also obtained.