The second inequality in (B.2) follows from the facts that for each j ∈ I$_{00}$ , P$_{1}$$_{j}$ and P$_{2}$$_{j}$ are independent, and Pr( P$_{ij}$ ≤ x ) ≤ x for all x ∈ [0 , 1] and i = 1 , 2 . The equality in (B.2) follows from the explanation of the equality in (A.3). We will now give an upper bound for each term of the sum in (B.1). First, Let us now prove that under the assumption that the p -values are independent, Pro- cedure 3.2 at levels ( q $^{′}$, 2 q $^{′}$) controls the FDR at level | I$_{00}$ | ( q $^{′}$) $^{2}$/m + ( | I$_{01}$ | /m + 1) q $^{′}$. Returning to the proof of Theorem 3.2, note that (A.1) can be rewritten as follows. B Proof for FDR control of the oracle Procedure 3.2 and the fact that for each j ∈ I$_{10}$, Pr( P$_{2}$$_{j}$ ≤ x ) ≤ x for all x ∈ [0 , 1] . The equality in (A.6) follows from the fact that ∪ |R$_{1}$ ( p$_{1}$ ) | r =1 C ( j ) r is a union of disjoint events, and Pr ( ∪ |R$_{1}$ ( p$_{1}$ ) | r =1 C ( j ) r | P$_{1}$ = p$_{1}$ ) = 1 . It follows from (A.6) that E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ / max( R, 1) ) ≤ q$_{2}$ . Using this fact and the bound (A.3) for the first term of (A.1), we obtain: FDR = E ( ∑ j ∈ $_{I$_{00}$}$R$_{j}$ max( R, 1) ) + E ( ∑ j ∈ $_{I$_{01}$}$R$_{j}$ max( R, 1) ) + E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ max( R, 1) ) . (B.1) E ( ∑ j ∈ $_{I$_{00}$}$R$_{j}$ max( R, 1) ) = ∑ j ∈ I$_{00}$ m ∑ r =1 1 r Pr ( j ∈ R $_{1}$, P$_{1}$$_{j}$ ≤ rq ′ m , P$_{2}$$_{j}$ ≤ rq ′ |R 1 | , C ( j ) r ) ≤ ∑ j ∈ I$_{00}$ m ∑ r =1 1 r Pr ( P$_{1}$$_{j}$ ≤ rq ′ m , P$_{2}$$_{j}$ ≤ q $^{′}$, C ( j ) r ) ≤ ( q $^{′}$) 2 m ∑ j ∈ I$_{00}$ m ∑ r =1 Pr ( C ( j ) r ) = | I$_{00}$ | m ( q $^{′}$) 2 (B.2) 31 FDR ≤ | I$_{0}$ | m q$_{1}$ + ( q − q$_{1}$ ) ≤ q$_{1}$ + ( q − q$_{1}$ ) = q.