Simes inequality ($\mathrm{Pr}({\min }_i{p}_{(i)}/i\le \alpha /m)\le \alpha$)

For $m\ge 1$ independent valid null p-values, the displayed inequality follows by applying the Benjamini-Hochberg procedure with all nulls true. Its false discovery rate then equals the probability of at least one rejection. Valid means $\mathrm{Pr}(p_i\le t)\le t$ for $0\le t\le 1$; independence is a sufficient hypothesis, and arbitrary dependence is not covered by this proof.

For the intersection of $m$ null hypotheses, reject when ${\min }_{i}m{p}_{(i)}/i\le \alpha$. The Simes inequality establishes a level-$\alpha$ test under independence of valid true-null p-values. The displayed quantity is its combined p-value.