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Optimal eigen expansions and uniform bounds. (English) Zbl 1349.62251

Summary: Let \(\{X_k\}_{k \in \mathbb {Z}} \in {\mathbb {L}}^2(\mathcal {T})\) be a stationary process with associated lag operators \(\boldsymbol {C}_h\). 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 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 \(\boldsymbol{G}\) in a general framework.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G70 Extreme value theory; extremal stochastic processes

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