the sum in (A.1): It follows from Lemma F.1, item 3, that the second term of the sum in (A.1) is bounded by q$_{2}$, hence Proof of item 2 of Theorem 3.3. We will first prove that the first term of the sum in (A.1) is bounded by q$_{1}$. For ˜ q$_{1}$ as defined in item 2 of Theorem 3.3, we denote k = ⌈ $^{tm/}$˜ q$_{1}$ − 1 ⌉ . The first term of the sum in (A.1) is upper bounded by two terms: m E ( ∑ j ∈ $_{I$_{0}$}$R$_{j}$ max( R, 1) ) = ∑ j ∈ I$_{0}$ m ∑ r =1 1 r Pr ( j ∈ R $_{1}$, P$_{1}$$_{j}$ ≤ $^{r}$˜ q$_{1}$ m , P$_{2}$$_{j}$ ≤ r ( q − q$_{1}$ ) |R 1 | , C ( j ) r ) ≤ j ∈ I$_{0}$ m r =1 1 r Pr P$_{1}$$_{j}$ ≤ min ( q$_{1}$ m , t ) , C ( j ) r ) (F.9) = ∑ j ∈ I$_{0}$ k ∑ r =1 1 r Pr ( P$_{1}$$_{j}$ ≤ $^{r}$˜ q$_{1}$ m , C ( j ) r ) + ∑ j ∈ I$_{0}$ m ∑ r = k +1 1 r Pr ( P$_{1}$$_{j}$ ≤ t, C ( j ) r ) , (F.10) ˜ p$_{jrl}$ = Pr ( P$_{1}$$_{j}$ ∈ ( ( l − $^{1)}$˜ q$_{1}$ m , $^{l}$˜ q$_{1}$ m ] , C ( j ) r ) . (F.11) where the inequality in (F.9) follows from the fact that j ∈ R 1 yields that P$_{1}$$_{j}$ ≤ t. We will now find an upper bound for each of the two terms in (F.10) separately. The derivation of the upper bound for the first term is along the lines of the derivation in the proof of item 1. We give it below for completeness. For each j ∈ I$_{0}$, r ∈ { 1 , . . . , m } , and l ∈ { 1 , . . . , m } , let us define: 42 E ( ∑ j ∈ $_{I$_{0}$}$R$_{j}$ max( R, 1) ) ≤ ∑ j ∈ I$_{0}$ q$_{1}$ m = | I$_{0}$ | q$_{1}$ m . (F.8) FDR ≤ | I$_{0}$ | q$_{1}$ m + q$_{2}$ = | I$_{0}$ | q$_{1}$ m + q − q$_{1}$ ≤ q.