tends to increase when the value of the Hurst parameter increases. Aside from that, the simulation results confirm the theoretical inference that long memory does not influence the asymptotic behavior of the CUSUM statistics when testing for a change in the mean in this situation. A comparison of the finite sample behavior of both CUSUM tests shows that: Therefore, it is not possible to definitely conclude that one of the tests outreaches the other. For the second scenario (when the ε$_{j}$ , j ≥ 1, are centered Pareto distributed), we considered all four testing procedures: CUSUM and self-normalized CUSUM; Wilcoxon and self-normalized Wilcoxon. In this case, it is notable that for all four testing procedures, heavier tails, i.e. a decrease of the tail parameter, go along with an decrease of rejections. A change in the Hurst parameter seems to have a remarkable effect on the finite sample performance of the Wilcoxon test only: An increase of dependence goes along with a decrease of rejections. • The self-normalized test has better size properties than the non-self-normalized test. • The empirical power of the non-self-normalized CUSUM test exceeds the empirical power of the self-normalized CUSUM test for most parameter combinations that have been considered. 25