 Solution

A procedure controls the familywise error rate at level $\alpha$ when, for every configuration of true nulls $I$,Let $i_{\ast }=\min I$. The step-down procedure can reject any true null only if it reaches and rejects $H_{i_{\ast }}$, which requires $p_{i_{\ast }}\le \alpha$. Since a valid p-value under its null satisfies $\mathbb{P}(p_{i_{\ast }}\le \alpha )\le \alpha$, the procedure controls FWER without any assumption on dependence among the p-values.

A distribution $P$ has the global Markov property for a directed acyclic graph $G$ when every D-separation statement $A{\perp }_{G}B\mid C$ implies the corresponding conditional independence $Z_A\perp \perp Z_B\mid Z_C$ under $P$.

If $G\in \mathcal{S}$ has nonadjacent vertices $j,k$, choose a topological ordering. Suppose $j$ precedes $k$. Then $j$ is a non-descendant and non-parent of $k$, so the directed local Markov property givesThe reversed ordering case is analogous. HenceUse the intersection-union test: reject $H_0$ exactly when $p_S\le \alpha$ for every $S$. Under $H_0$, at least one $H_{S_{\ast }}$ is true, soThis is nontrivial because it rejects whenever every tested conditional independence is rejected.