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## A Proofs 

In order to prove the STP of the rejection threshold, we first need to understand the the scaled inverse rejection process L ( t ) = t (max { R ( t ) , 1 } ) − 1 with t ∈ [0 , 1], with which the regularity of ˜ π$_{0}$ ( λ ), the property of the stopped backward filtration G and a contrapositive argument will yield Theorem 1. In what follows, no assumption will be made about the independence between the p-values, the continuity of the p-value distributions, or their stochastic orders wrt the standard uniform distribution. 

## A.1 Downward jumps of the scaled inverse rejection process 

Order the p-values into p$_{(1)}$ < p$_{(2)}$ < ... < p$_{(}$$_{n}$$_{)}$ distinctly, where the multiplicity of p$_{(}$$_{i}$$_{)}$ is n$_{i}$ for i = 1 , ..., n . Let p$_{(}$$_{n}$$_{+1)}$ = max { p$_{(}$$_{n}$$_{)}$, 1 } and p$_{(0)}$ = 0 . Define T$_{j}$ = j ∑ l =1 n$_{l}$ for j = 1 , ..., n . 

Lemma A.1. The process { L ( t ) , t ∈ [0 , 1] } is such that 

<img src='content_image/325488.jpg'>

Moreover, it can only be discontinuous at p$_{(}$$_{i}$$_{)}$ , 1 ≤ i ≤ n , where it can only have a downward jump with size 

Proof. Clearly 

and (A.1) holds. Therefore, the points of discontinuities of L ( · ) are the original distinct p-values. This justifies the first part of the assertion. 

<img src='content_image/325489.jpg'>

<img src='content_image/325490.jpg'>

