Asymptotic normality of the principal components of functional time series

We establish the asymptotic normality of the sample principal components of functional stochastic processes under nonrestrictive assumptions which admit nonlinear functional time series models. We show that the aforementioned asymptotic depends only on the asymptotic normality of the sample covariance operator, and that the latter condition holds for weakly dependent functional time series which admit expansions as Bernoulli shifts. The weak dependence is quantified by the condition of L4-m-approximability which includes all functional time series models in practical use. We also demonstrate convergence of the cross covariance operators of the sample functional principal components to their counterparts in the normal limit.