# Optimal eigen expansions and uniform bounds

arXiv: https://arxiv.org/abs/1501.01271 · HTML: https://mlsanity.com/paper/1501.01271/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jan 2015.

## Built on by (most cited first, 1 of 1)

- [Deep Spectral Q-learning with Application to Mobile Health](https://mlsanity.com/paper/2301.00927/) (2023)

## Abstract (arXiv)

Let biglXkbigr_k ∈ mathbbZ ∈ mathbbL2(T) be a stationary process with associated lag operators boldsymbolcal 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, 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 boldsymbolcal G in a general framework.