  To detect changes in the mean of a time series, one may use previsible
detection procedures based on nonparametric kernel prediction smoothers which
cover various classic detection statistics as special cases. Bandwidth
selection, particularly in a data-adaptive way, is a serious issue and not well
studied for detection problems. To ensure data adaptation, we select the
bandwidth by cross-validation, but in a sequential way leading to a functional
estimation approach. This article provides the asymptotic theory for the method
under fairly weak assumptions on the dependence structure of the error terms,
which cover, e.g., GARCH($p,q$) processes, by establishing (sequential)
functional central limit theorems for the cross-validation objective function
and the associated bandwidth selector. It turns out that the proof can be based
in a neat way on \cite{KurtzProtter1996}'s results on the weak convergence of
\ito integrals and a diagonal argument.
  Our gradual change-point model covers multiple change-points in that it
allows for a nonlinear regression function after the first change-point
possibly with further jumps and Lipschitz continuous between those
discontinuities.
  In applications, the time horizon where monitoring stops latest is often
determined by a random experiment, e.g. a first-exit stopping time applied to a
cumulated cost process or a risk measure, possibly stochastically dependent
from the monitored time series. Thus, we also study that case and establish
related limit theorems in the spirit of \citet{Anscombe1952}'s result. The
result has various applications including statistical parameter estimation and
monitoring financial investment strategies with risk-controlled early
termination, which are briefly discussed.
