The finite sample performance of the testing procedures is affected by a change of the tail parameter in the following way: lighter tails, i.e. an increase in the value of the tail parameter, go along with a decrease in power, but an empirical size which is closer to the level of significance. An increase of dependence, i.e. an increase of the Hurst parameter, results in a decrease of rejection rates. We make the following observations with respect to the behavior of the testing proce- dures: Due to the heavy size distortions of the non-self-normalized CUSUM test, it seems ad- visable to choose the self-normalized test over the non-self-normalized test in the case of heavy tails characterized by small values of α . However, for relatively light tails, the discrepancy between the test performances under the alternative suggests to choose the non-self-normalized test over the self-normalized CUSUM test. In all three test situations that have been considered, we observe that under the hy- pothesis the rejection rates of the self-normalized tests yield better approximations of the nominal level than the non-self-normalized tests. On the other hand, considering self- normalized tests instead of applying a deterministic normalization leads to a loss of power. In the context of change-point tests, the so-called “better size, less power” phenomenon has also been observed by Shao (2011), Shao and Zhang (2010) and Betken (2016b). More- over, it is consistent with observations made in Lobato (2001) and Sun et al. (2008). For a 33 • The non-self-normalized test suffers from size distortions in so far as the rejection rates under the hypothesis are considerably higher than the nominal level (especially for small values of α and H ). • All in all, the self-normalized test tends to be slightly undersized. • The non-self-normalized test has considerably better power than the self-normalized test for every parameter combination that has been considered.