content_image/295008.jpg content_image/295011.jpg content_image/295012.jpg content_image/295015.jpg
2. For Procedure 3.2 with parameters ( q$_{1}$, q ) , 

<img src='content_image/295008.jpg'>

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 ′ $_{1}$. 

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: 

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 } : 

<img src='content_image/295011.jpg'>

<img src='content_image/295012.jpg'>

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 } , 

<img src='content_image/295015.jpg'>

