  In this paper sequential monitoring schemes to detect nonparametric drifts
are studied for the random walk case. The procedure is based on a kernel
smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson
estimator and its as- sociated sequential partial sum process under
non-standard sampling. The asymptotic behavior differs substantially from the
stationary situation, if there is a unit root (random walk component). To
obtain meaningful asymptotic results we consider local nonpara- metric
alternatives for the drift component. It turns out that the rate of convergence
at which the drift vanishes determines whether the asymptotic properties of the
monitoring procedure are determined by a deterministic or random function.
Further, we provide a theoretical result about the optimal kernel for a given
alternative.
