Identifying the spectral representation of Hilbertian time series
- Citation:
- Horta, E., & Ziegelmann F. (2016). Identifying the spectral representation of Hilbertian time series. Statistics & Probability Letters. 118(November 2016), 45-49.
Abstract:
We provide √n-consistency results regarding estimation of the spectral representation of covariance operators of Hilbertian time series, in a setting with imperfect measurements. This is a generalization of the method developed in Bathia et al. (2010). The generalization relies on an important property of centered random elements in a separable Hilbert space, namely, that they lie almost surely in the closed linear span of the associated covariance operator. We provide a straightforward proof to this fact. This result is, to our knowledge, overlooked in the literature. It incidentally gives a rigorous formulation of Principal Component Analysis in Hilbert spaces.
