This paper investigates the problem of detecting relevant change points in the mean vector, say $μ_t =(μ_{1,t},\ldots ,μ_{d,t})^T$ of a high dimensional time series $(Z_t)_{t\in \mathbb{Z}}$. While the recent literature on testing for change points in this context considers hypotheses for the equality of the means $μ_h^{(1)}$ and $μ_h^{(2)}$ before and after the change points in the different components, we are interested in a null hypothesis of the form $$ H_0: |μ^{(1)}_{h} - μ^{(2)}_{h} | \leq Δ_h ~~~\mbox{ for all } ~~h=1,\ldots ,d $$ where $Δ_1, \ldots , Δ_d$ are given thresholds for which a smaller difference of the means in the $h$-th component is considered to be non-relevant. We propose a new test for this problem based on the maximum of squared and integrated CUSUM statistics and investigate its properties as the sample size $n$ and the dimension $d$ both converge to infinity. In particular, using Gaussian approximations for the maximum of a large number of dependent random variables, we show that on certain points of the boundary of the null hypothesis a standardised version of the maximum converges weakly to a Gumbel distribution.