A moment-based notion of time dependence for functional time series

This paper addresses the fundamental topic of time dependence for time series when data points are given as functions. We construct a notion of time dependence through the projections on the basis system extracted from the principal components of normalized sums. This allows us to adapt various scalar time series techniques to the functional data context. In particular, we define dependence based on the autocovariances and cumulants of the projections, covering short and long memory scenarios. This notion naturally applies to linear processes. We illustrate the applicability of this moment based approach through several statistical problems in functional time series: (i) investigating the consistency of the estimator of the functional principal components under short and long memory, (ii) estimating the long-run covariance function and (iii) testing for short memory against the long memory alternative.