  The paper is concerned with asymptotic properties of the principal components
analysis of functional data. The currently available results assume the
existence of the fourth moment. We develop analogous results in a setting which
does not require this assumption. Instead, we assume that the observed
functions are regularly varying. We derive the asymptotic distribution of the
sample covariance operator and of the sample functional principal components.
We obtain a number of results on the convergence of moments and almost sure
convergence. We apply the new theory to establish the consistency of the
regression operator in a functional linear model.
