content_image/295850.jpg content_image/295853.jpg
First, we find an upper bound for the first term of the sum in (A.1). 

<img src='content_image/295850.jpg'>

The equality in (A.2) follows from the independence of the p -values. The inequality in (A.3) follows from the fact that for each j ∈ I$_{0}$ , Pr( P$_{1}$$_{j}$ ≤ x ) ≤ x for all x ∈ [0 , 1] . Finally, the equality in (A.3) follows from the fact that ∪ m r $_{=1}$C ( j ) r is the entire sample space of ( P ( j ) 1 , P ( j ) 2 ) , represented as a union of disjoint events. 

Next, we find an upper bound for the second term of the sum in (A.1). Let R $_{1}$( p$_{1}$ ) be the set of selected indices using P$_{1}$ = p$_{1}$. Then E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ / max( R, 1) | P$_{1}$ = p$_{1}$ equals to: 

<img src='content_image/295853.jpg'>

The equality in (A.5) follows from the fact that P$_{2}$$_{j}$, P ( j ) 2 , P$_{1}$ are independent, since then C ( j ) r and the event { P$_{2}$$_{j}$ ≤ rq$_{2}$/ |R $_{1}$( p$_{1}$ ) |} are conditionally independent. The inequality in (A.6) follows from the independence of the p -values across the studies 

