SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\hat {f} (p) = \hat {f} \left(p ^ {(1)}, p ^ {(2)}\right) = \hat {g} \left(p ^ {(1)}\right) \cdot \hat {g} \left(p ^ {(2)}\right). \tag {A.44}
$$
By combining equations (A.13), (A.43), and (A.44), according to continuous mapping theorem [14], we have
$$
\sup  _ {p \in [ 0, 1 ]} \left| \widehat {\operatorname {f d r}} (p) - \operatorname {f d r} (p) \right| \xrightarrow {P} 0 \text {a s} n, J \rightarrow \infty . \tag {A.45}
$$
According to equation (A.45), we can verify the rationality of Assumption (i) in Theorem 2:
$$
\frac {1}{J} \sum_ {j = 1} ^ {J} \left| \widehat {\operatorname {f d r}} \left(p _ {j}\right) - \operatorname {f d r} \left(p _ {j}\right)\right| \xrightarrow {P} 0 \text {a s} n, J \rightarrow \infty . \tag {A.46}
$$
E.2.2 Proof of Theorem 2
To begin with, we introduce some notations. For  $\zeta \in (0,1]$ , denote
$$
\widehat {G} _ {J} ^ {0 0} (\zeta) = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {I} (p _ {j} \in \widehat {S} (\zeta)), \quad \widehat {G} _ {J} ^ {0 1} (\zeta) = \frac {1}{J _ {0 1}} \sum_ {j \in \Delta_ {0 1}} \mathbb {I} (p _ {j} \in \widehat {S} (\zeta)),
$$
$$
\widehat {G} _ {J} ^ {1 0} (\zeta) = \frac {1}{J _ {1 0}} \sum_ {j \in \Delta_ {1 0}} \mathbb {I} (p _ {j} \in \widehat {S} (\zeta)), \quad \widehat {G} _ {J} ^ {1 1} (\zeta) = \frac {1}{J _ {1 1}} \sum_ {j \in \Delta_ {1 1}} \mathbb {I} (p _ {j} \in \widehat {S} \zeta),
$$
$$
G _ {J} ^ {0 0} (\zeta) = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {P} \left(p _ {j} \in \widehat {S} (\zeta)\right), \quad G _ {J} ^ {0 1} (\zeta) = \frac {1}{J _ {0 1}} \sum_ {j \in \Delta_ {0 1}} \mathbb {P} \left(p _ {j} \in \widehat {S} (\zeta)\right),
$$
$$
G _ {J} ^ {1 0} (\zeta) = \frac {1}{J _ {1 0}} \sum_ {j \in \Delta_ {1 0}} \mathbb {P} \left(p _ {j} \in \widehat {S} (\zeta)\right),
$$
$$
\widehat {V} _ {J} ^ {0 0} (\zeta) = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {I} \left(p _ {j} \in \tilde {S} _ {0 0} (\zeta)\right), \quad \widehat {V} _ {J} ^ {0 1} (\zeta) = \frac {1}{J _ {0 1} + J _ {0 0}} \sum_ {j \in \Delta_ {0 1} \cup \Delta_ {0 0}} \mathbb {I} \left(p _ {j} \in \tilde {S} _ {0 1} (\zeta)\right),
$$
$$
\widehat {V} _ {J} ^ {1 0} (\zeta) = \frac {1}{J _ {1 0} + J _ {0 0}} \sum_ {j \in \Delta_ {1 0} \cup \Delta_ {0 0}} \mathbb {I} \left(p _ {j} \in \tilde {S} _ {1 0} (\zeta)\right),
$$
where  $J_{00} = |\Delta_{00}|$ ,  $J_{01} = |\Delta_{01}|$ ,  $J_{10} = |\Delta_{10}|$ ,  $J_{11} = |\Delta_{11}|$ . Denote  $r_J^{00} = J_{00} / J_{11}, r_J^{01} =$
$$
J _ {0 1} / J _ {1 1}, r _ {J} ^ {1 0} = J _ {1 0} / J _ {1 1}, v _ {J} = \frac {J _ {1 1}}{J _ {0 0} + J _ {0 1} + J _ {1 0}}. \text {A n d}
$$
$$
\overline {{K}} _ {J} ^ {0} (\zeta) = v _ {J} \left\{r _ {J} ^ {0 0} G _ {J} ^ {0 0} (\zeta) + r _ {J} ^ {0 1} G _ {J} ^ {0 1} (\zeta) + r _ {J} ^ {1 0} G _ {J} ^ {1 0} (\zeta) \right\},
$$
$$
K _ {J} ^ {0} (\zeta) = v _ {J} \{r _ {J} ^ {0 0} \widehat {G} _ {J} ^ {0 0} (\zeta) + r _ {J} ^ {0 1} \widehat {G} _ {J} ^ {0 1} (\zeta) + r _ {J} ^ {1 0} \widehat {G} _ {J} ^ {1 0} (\zeta) \},
$$
$$
\widehat {K} _ {J} ^ {0} (\zeta) = v _ {J} \{(r _ {J} ^ {0 1} + r _ {J} ^ {0 0}) \widehat {V} _ {J} ^ {0 1} (\zeta) + (r _ {J} ^ {1 0} + r _ {J} ^ {0 0}) \widehat {V} _ {J} ^ {1 0} (\zeta) - r _ {J} ^ {0 0} \widehat {V} _ {J} ^ {0 0} (\zeta) \},
$$
$$
\mathrm {F D P} _ {J} (\zeta) = \frac {r _ {J} ^ {0 0} \widehat {G} _ {J} ^ {0 0} (\zeta) + r _ {J} ^ {0 1} \widehat {G} _ {J} ^ {0 1} (\zeta) + r _ {J} ^ {1 0} \widehat {G} _ {J} ^ {1 0} (\zeta)}{r _ {J} ^ {0 0} \widehat {G} _ {J} ^ {0 0} (\zeta) + r _ {J} ^ {0 1} \widehat {G} _ {J} ^ {0 1} (\zeta) + r _ {J} ^ {1 0} \widehat {G} _ {J} ^ {1 0} (\zeta) + \widehat {G} _ {J} ^ {1 1} (\zeta)},
$$
$$
\mathrm {F D P} _ {J} ^ {\dagger} (\zeta) = \frac {\widehat {K} _ {J} ^ {0} (\zeta) / v _ {J}}{r _ {J} ^ {0 0} \widehat {G} _ {J} ^ {0 0} (\zeta) + r _ {J} ^ {0 1} \widehat {G} _ {J} ^ {0 1} (\zeta) + r _ {J} ^ {1 0} \widehat {G} _ {J} ^ {1 0} (\zeta) + \widehat {G} _ {J} ^ {1 1} (\zeta)},
$$
$$
\overline {{\mathrm {F D P}}} _ {J} (\zeta) = \frac {\overline {{K}} _ {J} ^ {0} (\zeta) / v _ {J}}{r _ {J} ^ {0 0} G _ {J} ^ {0 0} (\zeta) + r _ {J} ^ {0 1} G _ {J} ^ {0 1} (\zeta) + r _ {J} ^ {1 0} G _ {J} ^ {1 0} (\zeta) + \widehat {G} _ {J} ^ {1 1} (\zeta)}.
$$
Before proceeding with the proof of Theorem 2, we prove Lemma 2 first.
Lemma 2. Under Assumption (i)-(ii) in Theorem 2, if  $J_{00} \to \infty$ ,  $J_{01} \to \infty$ ,  $J_{10} \to \infty$  as  $J \to \infty$  and  $n \to \infty$ , we have in probability,
$$
\sup  _ {\zeta \in (0, 1 ]} \left| \widehat {G} _ {J} ^ {0 0} (\zeta) - G _ {J} ^ {0 0} (\zeta) \right| \longrightarrow 0, \quad \sup  _ {\zeta \in (0, 1 ]} \left| \widehat {G} _ {J} ^ {0 1} (\zeta) - G _ {J} ^ {0 1} (\zeta) \right| \longrightarrow 0,
$$
$$
\sup  _ {\zeta \in (0, 1 ]} \left| \widehat {G} _ {J} ^ {1 0} (\zeta) - G _ {J} ^ {1 0} (\zeta) \right| \longrightarrow 0, \quad \sup  _ {\zeta \in (0, 1 ]} \left| \widehat {K} _ {J} ^ {0} (\zeta) - \overline {{K}} _ {J} ^ {0} (\zeta) \right| \longrightarrow 0.
$$
Proof of Lemma 2. We consider three cases under composite null hypothesis.
Case 1 Under  $H_{00}$ : We can derive that
$$
\widehat {G} _ {J} ^ {0 0} (\zeta) = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left\{\mathbb {I} (p _ {j} \in \widehat {S} (\zeta)) - \mathbb {I} (p _ {j} \in S (\zeta)) + \mathbb {I} (p _ {j} \in S (\zeta)) \right\}, \tag {A.47}
$$
$$
G _ {J} ^ {0 0} (\zeta) = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left\{\mathbb {P} \left(p _ {j} \in \widehat {S} (\zeta)\right) - \mathbb {P} \left(p _ {j} \in S (\zeta)\right) + \mathbb {P} \left(p _ {j} \in S (\zeta)\right) \right\}.
$$
$$
\begin{array}{l} \sup  _ {\zeta \in (0, 1 ]} \left| \widehat {G} _ {J} ^ {0 0} (\zeta) - G _ {J} ^ {0 0} (\zeta) \right| \leq \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {I} \left(p _ {j} \in \widehat {S} (\zeta)\right) - \mathbb {I} \left(p _ {j} \in S (\zeta)\right) \right| \\ + \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} \left(p _ {j} \in \widehat {S} (\zeta)\right) - \mathbb {P} \left(p _ {j} \in S (\zeta)\right) \right| \tag {A.48} \\ + \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {I} \left(p _ {j} \in S (\zeta)\right) - \mathbb {P} \left(p _ {j} \in S (\zeta)\right) \right|. \\ \end{array}
$$
To deal with the first term, we have
$$
\begin{array}{l} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {I} (p _ {j} \in \widehat {S} (\zeta)) - \mathbb {I} (p _ {j} \in S (\zeta)) \right| \\ = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {I} \left\{\widehat {\operatorname {f d r}} (p _ {j}) \leq \zeta \right\} - \mathbb {I} \left\{\operatorname {f d r} (p _ {j}) \leq \zeta \right\} \right| \\ = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left[ \mathbb {I} \left\{\widehat {\mathrm {f d r}} (p _ {j}) \leq \zeta , \mathrm {f d r} (p _ {j}) > \zeta \right\} + \mathbb {I} \left\{\mathrm {f d r} (p _ {j}) \leq \zeta , \widehat {\mathrm {f d r}} (p _ {j}) > \zeta \right\} \right] \\ = \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left[ \mathbb {I} \left\{\widehat {\mathrm {f d r}} (p _ {j}) \leq \zeta , \zeta + \epsilon \geq \mathrm {f d r} (p _ {j}) > \zeta \right\} + \mathbb {I} \left\{\zeta - \epsilon <   \mathrm {f d r} (p _ {j}) \leq \zeta , \widehat {\mathrm {f d r}} (p _ {j}) > \zeta \right\} \right] \\ + \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left[ \mathbb {I} \left\{\widehat {\mathrm {f d r}} (p _ {j}) \leq \zeta , \mathrm {f d r} (p _ {j}) > \zeta + \epsilon \right\} + \mathbb {I} \left\{\mathrm {f d r} (p _ {j}) \leq \zeta - \epsilon , \widehat {\mathrm {f d r}} (p _ {j}) > \zeta \right\} \right] \\ \leq \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {I} \left\{\zeta - \epsilon <   \operatorname {f d r} \left(p _ {j}\right) \leq \zeta + \epsilon \right\} + \frac {1}{J _ {0 0} \epsilon} \sum_ {j \in \Delta_ {0 0}} \left| \widehat {\operatorname {f d r}} \left(p _ {j}\right) - \operatorname {f d r} \left(p _ {j}\right) \right|. \tag {A.49} \\ \end{array}
$$
Combine with the Glivenko-Cantelli theorem and Assumption (i), we can derive
$$
\begin{array}{l} Q := \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {I} \left\{\widehat {\operatorname {f d r}} \left(p _ {j}\right) \leq \zeta \right\} - \mathbb {I} \left\{\operatorname {f d r} \left(p _ {j}\right) \leq \zeta \right\} \right| \\ \leq \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {I} \left\{\zeta - \epsilon <   \operatorname {f d r} \left(p _ {j}\right) \leq \zeta + \epsilon \right\} + \frac {1}{J _ {0 0} \epsilon} \sum_ {j \in \Delta_ {0 0}} \left| \widehat {\operatorname {f d r}} \left(p _ {j}\right) - \operatorname {f d r} \left(p _ {j}\right) \right| \\ \leq \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} (\zeta - \epsilon <   \operatorname {f d r} (p _ {j}) \leq \zeta + \epsilon) \right| \\ + 2 \sup  _ {\zeta \in (0, 1 ]} \left| \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {I} \left\{\operatorname {f d r} \left(p _ {j}\right) \leq \zeta \right\} - \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \mathbb {P} \left\{\operatorname {f d r} \left(p _ {j}\right) \leq \zeta \right\} \right| \\ + \frac {1}{J _ {0 0} \epsilon} \sum_ {j \in \Delta_ {0 0}} \left| \widehat {\mathrm {f d r}} (p _ {j}) - \mathrm {f d r} (p _ {j}) \right| \\ \leq \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} (\zeta - \epsilon <   \operatorname {f d r} (p _ {j}) \leq \zeta + \epsilon) \right| + o _ {p} (1). \\ \end{array}
$$
Since  $\epsilon$  can be arbitrarily small,  $\sup_{\zeta \in (0,1]}\frac{1}{J_{00}}\sum_{j\in \Delta_{00}}|\mathbb{P}(\zeta -\epsilon <  \mathrm{fdr}(p_j)\leq \zeta +\epsilon)|$  can be small due to Assumption (ii). Consequently, we have  $Q = o_{p}(1)$  and thus the first term holds.
Before addressing the second term, we obtain that
$$
\begin{array}{l} \mathbb {P} \left(\widehat {\mathrm {f d r}} (p _ {j}) \leq \zeta\right) \\ \leq \mathbb {P} \left(\widehat {\mathrm {f d r}} (p _ {j}) \leq \zeta , \mathrm {f d r} (p _ {j}) \leq \zeta + \epsilon\right) + \mathbb {P} \left(\widehat {\mathrm {f d r}} (p _ {j}) \leq \zeta , \mathrm {f d r} (p _ {j}) > \zeta + \epsilon\right) \tag {A.50} \\ \leq \mathbb {P} \left(\operatorname {f d r} \left(p _ {j}\right) \leq \zeta + \epsilon\right) + \mathbb {P} \left(\left| \widehat {\operatorname {f d r}} \left(p _ {j}\right) - \operatorname {f d r} \left(p _ {j}\right) \right| > \epsilon\right). \\ \end{array}
$$
Combine with Assumption (i), we can derive that  $\mathbb{P}\left(\left|\widehat{\mathrm{fdr}}(p_j) - \mathrm{fdr}(p_j)\right| > \epsilon\right) \to 0$ .
$$
\begin{array}{l} \sup  _ {\zeta \in (0, 1 ]} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} \left(p _ {j} \in \widehat {S} (\zeta)\right) - \mathbb {P} \left(p _ {j} \in S (\zeta)\right) \right| \\ = \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} \left(\widehat {\operatorname {f d r}} \left(p _ {j}\right) \leq \zeta\right) - \mathbb {P} \left(\operatorname {f d r} \left(p _ {j}\right) \leq \zeta\right) \right| \\ \leq \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} \left(\operatorname {f d r} \left(p _ {j}\right) \leq \zeta + \epsilon\right) - \mathbb {P} \left(\operatorname {f d r} \left(p _ {j}\right) \leq \zeta\right) \right| \tag {A.51} \\ = \sup  _ {\zeta \in (0, 1 ]} \frac {1}{J _ {0 0}} \sum_ {j \in \Delta_ {0 0}} \left| \mathbb {P} (\zeta <   \operatorname {f d r} (p _ {j}) \leq \zeta + \epsilon) \right|. \\ \end{array}
$$
As  $\epsilon$  can be arbitrarily small,  $\sup_{\zeta \in (0,1]}\frac{1}{J_{00}}\sum_{j\in \Delta_{00}}|\mathbb{P}(\zeta < \mathrm{fdr}(p_j)\leq \zeta +\epsilon)|\to 0.$
The third term can be proved using the Glivenko-Cantelli theorem. Thus, we have shown the proof of the first claim in Lemma 2.
Cases 2-3 Under  $H_{01}$  and  $H_{10}$ : Following the similar procedure in Case 1, we can conclude the proof of the second and third claims in Lemma 2.
Case 4 According to the symmetric property of  $\mathfrak{p}$ -values  $p_j$  for  $j \in \Delta_{00} \cup \Delta_{01} \cup \Delta_{10}$ , we follow the similar steps in Case 1, thus, we have
$$
\sup  _ {\zeta \in (0, 1 ]} \left| \widehat {K} _ {J} ^ {0} (\zeta) - \overline {{K}} _ {J} ^ {0} (\zeta) \right| \longrightarrow 0. \tag {A.52}
$$
This concludes the proof of the fourth claim.
Proof of Theorem 2. For any  $\epsilon \in (0,\alpha)$ , suppose there exists  $\zeta_{\alpha -\epsilon} > 0$ , then
$$
\mathbb {P} \left(\operatorname {F D P} \left(\zeta_ {\alpha - \epsilon}\right) \leq \alpha - \epsilon\right)\rightarrow 1.
$$
By Lemma 2, for any constant  $c > 0$ , we have
$$
\sup  _ {0 <   \zeta \leq c} \left| \mathrm {F D P} _ {J} ^ {\dagger} (\zeta) - \mathrm {F D P} _ {J} (\zeta) \right| \xrightarrow {p} 0. \tag {A.53}
$$
By the definition of  $\zeta^{\star}$ , i.e.,  $\zeta^{\star} = \sup \left\{\zeta \in (0,1]:\mathrm{FDP}_{J}^{\dagger}(\zeta)\leq \alpha \right\}$ , we have
$$
\begin{array}{l} \mathbb {P} \left(\zeta^ {\star} \geq \zeta_ {\alpha - \epsilon}\right) \geq \mathbb {P} \left(\mathrm {F D P} _ {J} ^ {\dagger} \left(\zeta_ {\alpha - \epsilon}\right) \leq \alpha\right) \\ \geq \mathbb {P} \left(\left| \mathrm {F D P} _ {J} ^ {\dagger} (\zeta_ {\alpha - \epsilon}) - \mathrm {F D P} _ {J} (\zeta_ {\alpha - \epsilon}) \right| \leq \epsilon , \mathrm {F D P} (\zeta_ {\alpha - \epsilon}) \leq \alpha - \epsilon\right) \\ = \mathbb {P} \left(\mathrm {F D P} \left(\zeta_ {\alpha - \epsilon}\right) \leq \alpha - \epsilon\right) - \mathbb {P} \left(\left| \mathrm {F D P} _ {J} ^ {\dagger} \left(\zeta_ {\alpha - \epsilon}\right) - \mathrm {F D P} _ {J} \left(\zeta_ {\alpha - \epsilon}\right) \right| > \epsilon , \mathrm {F D P} \left(\zeta_ {\alpha - \epsilon}\right) \leq \alpha - \epsilon\right) \\ \geq \mathbb {P} \left(\operatorname {F D P} \left(\zeta_ {\alpha - \epsilon}\right) \leq \alpha - \epsilon\right) - \mathbb {P} \left(\left| \operatorname {F D P} _ {J} ^ {\dagger} \left(\zeta_ {\alpha - \epsilon}\right) - \operatorname {F D P} _ {J} \left(\zeta_ {\alpha - \epsilon}\right) \right| > \epsilon\right) \\ \geq 1 - \epsilon , \tag {A.54} \\ \end{array}
$$
for  $J$  large enough. Thus, we have
$$
\mathbb {P} \left(\zeta^ {\star} \geq \zeta_ {\alpha - \epsilon}\right) \geq 1 - \epsilon . \tag {A.55}
$$
Conditioning on the event  $\zeta^{\star} \geq \zeta_{\alpha - \epsilon}$ , we have
$$
\begin{array}{l} \limsup_{n,J\to \infty}\mathbb{E}\left[ \mathrm{FDP}_{J}\left(\zeta^{\star}\right)\right]\leq \limsup_{n,J\to \infty}\mathbb{E}\left[ \mathrm{FDP}_{J}\left(\zeta^{\star}\right)\mid \zeta^{\star}\geq \zeta_{\alpha -\epsilon}\right]\mathbb{P}\left(\zeta^{\star}\geq \zeta_{\alpha -\epsilon}\right) + \epsilon \\ \leq \limsup_{n,J\to \infty}\mathbb{E}\left[ \left|FDP_{J}(\zeta^{\star}) - \overline{FDP}_{J}(\zeta^{\star})\right|\mid \zeta^{\star}\geq \zeta_{\alpha -\epsilon}\right]\mathbb{P}(\zeta^{\star}\geq \zeta_{\alpha -\epsilon}) \\ + \lim  _ {n, J \rightarrow \infty} \sup  _ {\zeta^ {\star}} \mathbb {E} \left[\left| \mathrm {F D P} _ {J} ^ {\dagger} (\zeta^ {\star}) - \overline {{\mathrm {F D P}}} _ {J} (\zeta^ {\star}) \right| | \zeta^ {\star} \geq \zeta_ {\alpha - \epsilon} \right] \mathbb {P} (\zeta^ {\star} \geq \zeta_ {\alpha - \epsilon}) \\ + \lim  _ {n, J \rightarrow \infty} \sup  _ {\zeta \rightarrow \infty} \mathbb {E} \left[ \mathrm {F D P} _ {J} ^ {\dagger} (\zeta^ {\star}) \mid \zeta^ {\star} \geq \zeta_ {\alpha - \epsilon} \right] \mathbb {P} (\zeta^ {\star} \geq \zeta_ {\alpha - \epsilon}) + \epsilon \\ \leq \limsup_{n,J\to \infty}\mathbb{E}\left[\sup_{\zeta \in [\zeta_{\alpha -\epsilon},1]}\bigl|\mathrm{FDP}_{J}(\zeta) - \overline{\mathrm{FDP}}_{J}(\zeta)\bigr|\right] \\ + \lim  _ {n, J \rightarrow \infty} \mathbb {E} \left[ \sup  _ {\zeta \in [ \zeta_ {\alpha - \epsilon}, 1 ]} \left| \mathrm {F D P} _ {J} ^ {\dagger} (\zeta) - \overline {{\mathrm {F D P}}} _ {J} (\zeta) \right|\right] \\ + \limsup_{n,J\to \infty}\mathbb{E}\left[ \mathrm{FDP}^{\dagger}_{J}\left(\zeta^{\star}\right)\right] + \epsilon . \\ \end{array}
$$
(A.56)
The first two terms are 0 based on Lemma 2 and the dominated convergence theorem. For the third term, we have  $\mathrm{FDP}_J^\dagger (\zeta^\star)\leq \alpha$  by the definition of  $\zeta^{\star}$ . This concludes the proof of Theorem 2.