  We study the problem of change point localisation and inference for
sequentially collected fragmented functional data, where each curve is observed
only over discrete grids randomly sampled over a short fragment. The sequence
of underlying covariance functions is assumed to be piecewise constant, with
changes happening at unknown time points. To localise the change points, we
propose a computationally efficient fragmented functional dynamic programming
(FFDP) algorithm with consistent change point localisation rates. With an extra
step of local refinement, we derive the limiting distributions for the refined
change point estimators in two different regimes where the minimal jump size
vanishes and where it remains constant as the sample size diverges. Such
results are the first time seen in the fragmented functional data literature.
As a byproduct of independent interest, we also present a non-asymptotic result
on the estimation error of the covariance function estimators over intervals
with change points inspired by Lin et al. (2021). Our result accounts for the
effects of the sampling grid size within each fragment under novel
identifiability conditions. Extensive numerical studies are also provided to
support our theoretical results.
