Suppose I have $p$ null hypotheses $H_1,\ldots,H_p$ and the global hypothesis: $$ H_0:H_1 \cap H_2 \cap \cdots \cap H_p. $$ I'm interested in level controlling when testing for $H_0$ (using multiple testing): bounding $$ \Pr(\text{reject }H_0|H_0\text{ is true}) $$ regardless of the underlying dependence of the individual tests used to test $H_1,\ldots,H_p$. As this wiki article explains, I am in fact interested in controlling the family-wise error rate (FWER) in the weak sense.
The same article lists 2 methods of FWER level control (Bonferroni and Holm) that are robust to dependence of the individual tests. But these control FWER in the strong sense (defined in the article). So they work for my purpose but I'm afraid they are "overcompensating." What are some (modern) references for FWER level control in the weak sense?
According to Remark 3.2 from [1], FDR coincides with weak FWER:
Given that all null hypotheses are null, any discovery is a false discovery, and therefore, $$FDP = V/R = 1.$$ ($V$ = # of false discoveries, $R$ = # of discoveries)
As you can see, under the condition that all nulls are true, the definitions of Type I error of a multiple test and FDP already coincide.
Since $FDR = E(FDP)$, when FDR is controlled at level $\alpha$, the Type I error given all nulls are true is controlled at level $\alpha$ as well.
Thus, FDR is a good choice for your problem.
[1] Rosenblatt, Jonathan: A Practitioner's Guide to Multiple Testing Error Rates. Available online at http://arxiv.org/pdf/1304.4920v3