A Appendix

It suggests that, for any m { k,...,n 1 } and z R, L A.2 Proofs for Lemma 2 and 3 for the case when K is unknown in 4 Lemma 2 . It suggests that, for any m { 0,...,n 1 } and z R, L For any m { 0,...,n 1 } and z R, we have L A.3 Additional tricks for methods proposed in 3. Finding optimal CP vector when z = in paraCP(n,k, ˆ T Additional pruning condition for parametric DP when K is fixed. In 3.3, we showed that Lemma 4. F orn [ N ], and k [ K ], let T Therefore, it fails to control the false positive rate. This is asymptotic test for multiple detected CPs. Fused Lasso (proposed by the same authors), is worse than BinSeg-SI. BinSeg-SI had been considered as a computationally efficient approximation of the problem in (7), where the authors additionally condition on extra information for computational tractability, e.g., the order that CPs are detected.