Optimal eigen expansions and uniform bounds
Optimal eigen expansions and uniform bounds
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DOI：
10.1007/s00440-015-0671-3
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发表时间：
2015-11
影响因子：
2
通讯作者：
M. Jirak
中科院分区：
文献类型：
--
作者：
M. Jirak
设是带有滞后算子的平稳过程$${\varvec{\mathcal{C}}_h$$。在几乎最优的条件下，建立了滞后算子的经验特征值和特征函数的一致渐近展开式。此外，潜在的相关性假设在一定意义上是最优的，包括短记忆过程和长记忆过程。这使我们能够研究在非常一般的条件下经验本征值的相对最大偏差。此外，还显示了收敛到极值分布的情况。在一般框架下，我们还讨论了渐近展开式如何转化为长期协方差算子$$\varvec{\mathcal{G}}$$。
Letbe a stationary process with associated lag operators $${\varvec{\mathcal {C}}}_h$$. 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 $${\varvec{\mathcal {G}}}$$ in a general framework.