2. For Procedure 3.2 with parameters ( q$_{1}$, q ) , 3. Item 2 holds if in the terms rq$_{1}$/m and R$_{2}$q$_{1}$/m in step 2 of Procedure 3.2, q$_{1}$ is replaced by q ′ $_{1}$, for any value of q ′ $_{1}$. E ( ∑ j ∈ I$_{10}$ R$_{j}$ max( R, 1) ) ≤ q$_{2}$. See Section F.1 for a proof. Proof of item 1 of Theorem 3.3. We will first show that the first term of the sum in (A.1) is bounded by | I$_{0}$ | q$_{1}$/m. We will use the technique developed in Benjamini and Yekutieli (2001) in the proof of their Theorem 1.3. For each j ∈ I$_{0}$, r ∈ { 1 , . . . , m } , and l ∈ { 1 , . . . , m } , let us define: p$_{jrl}$ = Pr ( P$_{1}$$_{j}$ ∈ ( ( l − 1) q$_{1}$ m ∑ m s =1 1 s , lq$_{1}$ m ∑ m s =1 1 s ] , C ( j ) r ) . m ∑ r =1 p$_{jrl}$ = Pr ( P$_{1}$$_{j}$ ∈ ( ( l − 1) q$_{1}$ m ∑ m s =1 1 s , lq$_{1}$ m ∑ m s =1 1 s ] , ∪ m r $_{=1}$C ( j ) r ) = Pr P$_{1}$$_{j}$ ∈ ( l − 1) q$_{1}$ m m s =1 1 s , lq$_{1}$ m m s =1 1 s ]) . (F.1) Since ∪ m r $_{=1}$C ( j ) r is the entire sample space represented as a union of disjoint events, we obtain for each j ∈ I$_{0}$ and l ∈ { 1 , . . . , m } : Note that for j ∈ I$_{0}$, Pr ( P$_{1}$$_{j}$ ≤ x ) ≤ x for all x ≥ 0, in particular Pr ( P$_{1}$$_{j}$ = 0) = 0 . Therefore, for each j ∈ I$_{0}$ and r ∈ { 1 , . . . , m } , Pr ( P$_{1}$$_{j}$ ≤ rq$_{1}$ m ∑ m s =1 1 s , C ( j ) r ) = r ∑ l =1 p$_{jrl}$. (F.2) 40