  This paper considers the problem of testing if a sequence of means
$(\mu_t)_{t =1,\ldots ,n }$ of a non-stationary time series $(X_t)_{t =1,\ldots
,n }$ is stable in the sense that the difference of the means $\mu_1$ and
$\mu_t$ between the initial time $t=1$ and any other time is smaller than a
given level, that is $ | \mu_1 - \mu_t | \leq c $ for all $t =1,\ldots ,n $. A
test for hypotheses of this type is developed using a biascorrected monotone
rearranged local linear estimator and asymptotic normality of the corresponding
test statistic is established. As the asymptotic variance depends on the
location and order of the critical roots of the equation $| \mu_1 - \mu_t | =
c$ a new bootstrap procedure is proposed to obtain critical values and its
consistency is established. As a consequence we are able to quantitatively
describe relevant deviations of a non-stationary sequence from its initial
value. The results are illustrated by means of a simulation study and by
analyzing data examples.
