No Universal Multiplicative FDR Bound for the Benjamini-Hochberg Procedure with Correlated Two-Sided Gaussian Tests

We study the worst-case false discovery rate (FDR) of the Benjamini–Hochberg (BH) procedure applied to two-sided Gaussian $p$-values under unrestricted correlation. Although BH need not control the FDR exactly at its nominal level $q$ in this setting, it has been conjectured that its FDR is bounded by $Cq$ for some universal constant $C$. We disprove this conjecture. For every $q\in (0,1)$, we construct finite-dimensional Gaussian models with positive definite correlation matrices and positive correlations whose FDR approaches an explicit lower bound $\mathrm{\ell }(q)>q$. As $q\downarrow 0$, $$\mathrm{\ell }(q)=\frac{q\sqrt{\log (1\mathrm{/}q)}}{2\sqrt{\pi }}+c_{\mathrm{\ell }}q+o(q),c_{\mathrm{\ell }}=0.6492828\dots \mathrm{.}$$ Consequently, the worst-case inflation factor $\mathrm{FDR}({\mathrm{B}\mathrm{H}}_q)\mathrm{/}q$ diverges at least on the order of $\sqrt{\log (1\mathrm{/}q)}$. We also establish the matching upper bound $O\left(q\sqrt{\log (1\mathrm{/}q)}\right)$, uniformly over a broad class of common-factor Gaussian models with arbitrary means and loadings. Thus, the rate $q\sqrt{\log (1\mathrm{/}q)}$ is sharp within this class.