On the false discovery proportion convergence under Gaussian equi-correlation

We study the convergence of the false discovery proportion (FDP) of the Benjamini-Hochberg procedure in the Gaussian equi-correlated model, when the correlation ${\rho }_m$ converges to zero as the hypothesis number $m$ grows to infinity. By contrast with the standard convergence rate $m^{1\mathrm{/}2}$ holding under independence, this study shows that the FDP converges to the false discovery rate (FDR) at rate $\{\min (m,1\mathrm{/}{\rho }_m){\}}^{1\mathrm{/}2}$ in this equi-correlated model.