2016/04/10 by Horta, Eduardo, Ziegelmann, Flavio
#60G10 #62G99 #62M99 #FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1604.02702
We provide square-root 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 PCA in Hilbert spaces.