consideration, we may, as before, conclude that If we choose γ$_{n}$ , n ∈ N , such that converges to 0, the right-hand side of the above inequality vanishes as n tends to ∞ due to the choice of γ$_{n}$ . This concludes the proof of (S1.1) as well as the proof of Lemma 3.6. S2 Simulations We computed the empirical size and the empirical power of the testing procedures on the basis of simulated time series according to the corresponding specifications in Section 6 of the main document. In addition to the simulation results referred to in the main document, the simulation tables contain the empirical power of the testing procedures when the change point is located closer to the testing region ( τ = 0 . 25). The rejection rates are reported in Tables 5 to 10. P ( $_{ω}$′′ M ∗ $_{n}$( δ ) ≥ ε ) ≤ 2 K ε 2 $_{γ}$− 1 n 1 n n ∑ j =1 ( F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$( δ$_{n}$ ) − F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$(0) ) ≤ 2 K ε 2 $_{γ}$− 1 n E ( sup 0 ≤ x ≤ 1 ( F$_{n}$ ( δ$_{n}$ + x ) − F$_{n}$ ( x )) ) . $_{γ}$− 1 n E ( sup 0 ≤ x ≤ 1 ( F$_{n}$ ( δ$_{n}$ + x ) − F$_{n}$ ( x )) ) 46