  Functional principal components (FPC's) provide the most important and most
extensively used tool for dimension reduction and inference for functional
data. The selection of the number, d, of the FPC's to be used in a specific
procedure has attracted a fair amount of attention, and a number of reasonably
effective approaches exist. Intuitively, they assume that the functional data
can be sufficiently well approximated by a projection onto a finite-dimensional
subspace, and the error resulting from such an approximation does not impact
the conclusions. This has been shown to be a very effective approach, but it is
desirable to understand the behavior of many inferential procedures by
considering the projections on subspaces spanned by an increasing number of the
FPC's. Such an approach reflects more fully the infinite-dimensional nature of
functional data, and allows to derive procedures which are fairly insensitive
to the selection of d. This is accomplished by considering limits as d tends to
infinity with the sample size.
  We propose a specific framework in which we let d tend to infinity by
deriving a normal approximation for the two-parameter partial sum process of
the scores \xi_{i,j} of the i-th function with respect to the j-th FPC. Our
approximation can be used to derive statistics that use segments of
observations and segments of the FPC's. We apply our general results to derive
two inferential procedures for the mean function: a change-point test and a
two-sample test. In addition to the asymptotic theory, the tests are assessed
through a small simulation study and a data example.
