The first inequality in (F.22) follows from the independence of the p -values across the studies and the fact that for each j ∈ I$_{10}$, Pr( P$_{2}$$_{j}$ ≤ x ) ≤ x for all x ≥ 0 . The second inequality in (F.22) follows from Lemma F.1, item 1. Taking the expectation over P$_{1}$, we obtain E ( ∑ j ∈ $_{I$_{01}$}$R$_{j}$ / max( R, 1) ) ≤ q$_{2}$. Proof of item 3. For q ′ 1 and p$_{1}$ arbitrary fixed, The arguments that lead from (F.21) to the result of item 2 complete the proof. Theorem G.1. Assume that the p -values across studies are independent, the p -values within the primary study are independent, and the set of p -values within the follow-up study has property PRDS. If the selection rule used in step 1 of Procedure 3.2 is a valid selection rule, then Procedure 3.2 with parameters ( q$_{1}$, q ) controls the FDR at level q for the family of no replicability null hypotheses H$_{NR,}$$_{1}$, . . . , H$_{NR,m}$ . Proof. Let us first find an upper bound for the first term of the sum in (A.1). Note that (A.3) is established using the independence of the p -values within the primary study only, therefore it holds for any form of dependence among the p -values within the follow-up study. In particular, (A.3) holds under the dependency of Theorem G.1, establishing an upper bound for the first term of the sum in (A.1). It follows 47 G Additional theoretical results under dependence E ( ∑ j ∈ I$_{10}$ R$_{j}$/ max( R, 1) | P$_{1}$ = p$_{1}$ ) = j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 1 r I [ p$_{1}$$_{j}$ ≤ rq ′ 1 m ] Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , C ( j ) r | P$_{1}$ = p$_{1}$ ) ≤ ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 1 r Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , C ( j ) r | P$_{1}$ = p$_{1}$ ) .