Now we show that L ( · ) can only have downward jumps at points of discontinuity. Define From the fact R ( p$_{(}$$_{j}$$_{)}$ ) − R ( p$_{(}$$_{j}$$_{)}$ − ) = n$_{j}$ > 0 but R ( p$_{(}$$_{j}$$_{)}$ + ) − R ( p$_{(}$$_{j}$$_{)}$ ) = 0 for each 1 ≤ j ≤ n, it follows that ϕ ( p$_{(}$$_{j}$$_{)}$, 0+ ) = lim$_{η}$$_{↓}$$_{0}$ ϕ ( t, η ) = 0 but Thus L ( p$_{(}$$_{j}$$_{)}$ − ) − L ( p$_{(}$$_{j}$$_{)}$ ) = ϕ ( p$_{(}$$_{j}$$_{)}$, 0 − ) > 0 and the proof is completed. Lemma A.1 shows that the process L ( t ) is piecewise linear and can only have downward jumps as t increases. The conclusion of Lemma A.1 is right the contrary to the claim in the proof of Theorem 2 in Storey et al. (2004) that “the process mt/R ( t ) has only upward jumps and has a final value of 1”, since it says “the process mt/R ( t ) has only downward jumps”. We construct a counterexample to their claim as follows. For a small increase c in t which results an increase a$_{c}$ in R ( t ), we see that if and only if c a$_{c}$ < t R ( t ) . Construct m p-values with n ≥ 4 such that there exists some 1 ≤ j$_{0}$ < n − 2 with n$_{j}$$_{0}$$_{+1}$ > T$_{j}$$_{0}$ but p$_{(}$$_{j}$$_{0}$$_{+1)}$ < 1. Choose c$_{1}$ and c$_{2}$ such that 0 < c$_{1}$ < ( p$_{(}$$_{j}$$_{0}$$_{+1)}$ − p$_{(}$$_{j}$$_{0}$$_{)}$ ) / 2 and 0 < c$_{2}$ < p$_{(}$$_{j}$$_{0}$$_{+1)}$ − 2 c$_{1}$ . Let t$_{0}$ = p$_{(}$$_{j}$$_{0}$$_{+1)}$ − c$_{1}$ and c = c$_{1}$ + c$_{2}$ . Then p$_{(}$$_{j}$$_{0}$$_{)}$ < t$_{0}$ < p$_{(}$$_{j}$$_{0}$$_{+1)}$ , R ( t$_{0}$ ) = T$_{j}$$_{0}$ and R ( t$_{0}$ + c ) = T$_{j}$$_{0}$ + n$_{j}$$_{0}$$_{+1}$ . Further, 0 < c < t$_{0}$ and a$_{c}$ = n$_{j}$$_{0}$$_{+1}$ . So c n$_{j}$$_{0}$$_{+1}$ < t$_{0}$ T$_{j}$$_{0}$ and L ( t$_{0}$ + c ) − L ( t$_{0}$ ) < 0. Letting c → 0 gives p$_{(}$$_{j}$$_{0}$$_{+1)}$ as a point of downward jump for L ( t ). When α = 0, ˜ t$_{α}$ ( λ ) = 0 if ˜ π$_{0}$ ( λ ) > 0 but ˜ t$_{α}$ ( λ ) = 1 if ˜ π$_{0}$ ( λ ) = 0. In this case, ˜ t$_{α}$ ( λ ) is already a stopping time and ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ( λ ) = α = 0. On the other hand, ˜ t$_{α}$ ( λ ) = 1 when α = 1. In this case, ˜ t$_{α}$ ( λ ) is a stopping time and ˜ FDR$_{λ}$ ( ˜ t$_{α}$ ( λ ) = ˜ π$_{0}$ ( λ ). Note that ˜ π$_{0}$ ( λ ) = 0 can not happen when 0 < ˜ t$_{α}$ ( λ ) < 1. A.2 Proof of Theorem 1 8 ϕ ( p$_{(}$$_{j}$$_{)}$, 0 − ) = lim η ↑ 0 ϕ ( p$_{(}$$_{j}$$_{)}$, η ) = p$_{(}$$_{j}$$_{)}$n$_{j}$ R ( p$_{(}$$_{j}$$_{)}$ ) [ R ( p$_{(}$$_{j}$$_{)}$ ) − n$_{j}$ ] > 0. ϕ ( t, η ) = t + η R ( t + η ) − t R ( t ) = ηR ( t ) + t [ R ( t ) − R ( t + η )] R ( t + η ) R ( t ) . L ( t + c ) − L ( t ) = t + c R ( t ) + a$_{c}$ − t R ( t ) = cR ( t ) − ta$_{c}$ [ R ( t ) + a$_{c}$ ] R ( t ) < 0