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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 agiven level, that is $ | \mu_1 - \mu_t | \leq c $ for all $t =1,\ldots ,n $. Atest for hypotheses of this type is developed using a biascorrected monotonerearranged local linear estimator and asymptotic normality of the correspondingtest statistic is established. As the asymptotic variance depends on thelocation 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 itsconsistency is established. As a consequence we are able to quantitativelydescribe relevant deviations of a non-stationary sequence from its initialvalue. The results are illustrated by means of a simulation study and byanalyzing data examples.

Change point analysis in non-stationary processes - a mass excess approach