Furthermore, I show that it is innocuous to assume that b$_{k}$ + δ ≤ t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ b$_{k}$$_{+1}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − t$_{W}$ w at the optimum. If there exists an integer k s.t. one can always construct { ˜ B$_{k}$ } ∞ k = −∞ satisfying by defining ˜ B$_{k}$ = $^{(}$−∞ ,$_{˜}$ b$_{k}$ ) as follows: The inequality in (A.14) is illustrated in Figure A.7, which describes Figure. A.7: ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( B$_{k}$ | w ) − µ$_{1}$$_{,W}$ $^{(}$B D k | w $^{)}$, 0 $^{}}$≤ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( ˜ B$_{k}$ | w $^{)}$− µ$_{1}$$_{,W}$ ( ˜ B D k | w $^{)}$, 0 } 71 ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( B$_{k}$ | w ) − µ$_{1}$$_{,W}$ $^{(}$B D k | w $^{)}$, 0 $^{}}$, ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( ˜ B$_{k}$ | w $^{)}$− µ$_{1}$$_{,W}$ ( ˜ B D k | w $^{)}$, 0 } ˜ b$_{j}$ = b$_{j}$ for j ≤ k, ˜ b$_{k}$$_{+1}$ = t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{W}$ ( b$_{k}$ + δ ) + t$_{1}$ − t$_{0}$ t$_{1}$ − t$_{W}$ w, ˜ b$_{j}$$_{+1}$ = b$_{j}$ for j ≥ k + 1 . ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( B$_{k}$ | w ) − µ$_{1}$$_{,W}$ $^{(}$B D k | w $^{)}$, 0 } ≤ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( ˜ B$_{k}$ | w $^{)}$− µ$_{1}$$_{,W}$ ( ˜ B D k | w $^{)}$, 0 $^{}}$, (A.14) b$_{k}$ + δ > t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ b$_{k}$$_{+1}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − t$_{W}$ w (a) (b)