There was an error in the original publication: \"By taking infinite summations of \(p_{DSA,j}\) in Lemma 1 for all indices \(i\in\mathbb{N}\), we can compute the probability \(P_{DSA}\) that a DS attack will ever achieve the necessary conditions in Definition 1.\"A correction has been made to this sentence located in Section 3, The Attack Probabilities, in the paragraph 4 on line 1:  
\"By taking infinite summations of \(p_{DSA,i}\) in Lemma 1 for all indices \(i\in\mathbb{N}\), we can compute the probability \(\mathbb{P}_{DSA}\) that a DS attack will ever achieve the necessary conditions in Definition 1.\"  
There was an error in the original publication: \"If there does not exist such an index \(i\), with the probability of \(1-P_{DSA}\), then \(T_{DSA}=\infty\).\"  
A correction has been made to this sentence located in Section 3, The Attack Probabilities, in the paragraph 5 on line 2:  
\"If there does not exist such an index \(i\), with the probability of \(1-\mathbb{P}_{DSA}\), \(T_{DSA}=\infty\).\"  
There was an error to Equation (19) in the original publication:  
\[{}^{\omega}f_{T_{DSA}}(t)=\sum_{i=2N_{BC}+1}^{\infty}p_{DSA,i}f_{T_{i}}(t)+(1- P_{DSA})\delta(t-\infty),^{\omega}\]  
A correction has been made to Equation (19):  
\[{}^{\omega}f_{T_{DSA}}(t)=\sum_{i=2N_{BC}+1}^{\infty}p_{DSA,i}f_{T_{i}}(t)+(1- \mathbb{P}_{DSA})\delta(t-\infty),^{\omega}\]  
There was an error in the original publication: \"The previous probabilistic analyses in [1,6] show that the success of DS attacks is not guaranteed when \(p_{A}<0.5\).\"  
A correction has been made to this sentence located in Section 4, Profitable DS Attacks, in the paragraph 1 on line 1:  
\"The previous probabilistic analyses in [1,6] have shown that the success of DS attacks is not guaranteed when \(p_{A}<0.5\).\"  
There was an error in the original publication: \"With regards to \(P\), if an attack succeeds, the revenue comes from \(C\), as it is double-spent, and \(R\) for the number of blocks minded during the time duration \(T_{AS}\), i.e., \(R(\lambda_{A},T_{AS})\).\"  
A correction has been made to this sentence located in Section 4, Profitable DS Attacks, in the paragraph 4 on line 1:  
\"With regards to \(P\), if an attack succeeds, the revenue comes from \(C\), as it is double-spent, added to \(R\) for the number of blocks mined during the time duration \(T_{AS}\), i.e., \(R(\lambda_{A},T_{AS})\).\"  
There was an error to Equation (35) in the original publication:  
\[{}^{\omega}C_{\text{Req.}}==\frac{\mathbb{E}_{X}}{\mathbb{P}_{AS}}-\mathbb{E }[R(\lambda_{A},T_{AS})].^{\omega}\]  
A correction has been made to Equation (35):  
\[{}^{\omega}C_{\text{Req.}}=\frac{\mathbb{E}_{X}}{\mathbb{P}_{AS}}-\mathbb{E }[R(\lambda_{A},T_{AS})].^{\omega}\]  
There was an error in the original publication: \"Specifically, under, it is required for the honest chain to have added \(N_{BC}\) blocks, while under their condition, it was the fraudulent chain to do so (see Section 3 of [16]).\"  
A correction has been made to a sentence located in Section 6, Related Works, in the paragraph 10 on line 6:  
\"Specifically, under our definition, it is required for the honest chain to have added \(N_{BC}\) blocks, while under their condition it was fraudulent for the chain to do so (see Section 3 of [16]).\"There was an error in the original publication: \"Note that \(\mathcal{D}_{j}^{(1)}\cap\left(\mathcal{D}_{ij}^{(2)}\right)=\phi\) for \(i\leq 2N_{BC}\), since the minimum number of states for an attack success is \(2N_{BC}+1\); \(N_{BC}\) number of \(+1\)'s state transitions for the block confirmation; and \(N_{BC}+1\) number of \(-1\)'s state transitions for the success of PoW competition. Thus, \(p_{PDSA,i}=0\) for \(i\leq 2N_{BC}\).