Optimal FDR-FNR tradeoff beyond independent two-group mixtures

For each dimension $n\ge 1$, let $(X^{(n)},H^{(n)})$ be a random pair with $X^{(n)}=(X_1,\dots,X_n)\in\mathcal X^n$ observed and $H^{(n)}=(H_1,\dots,H_n)\in\{0,1\}^n$ latent, where $H_i=0$ means the $i$-th null hypothesis is true and $H_i=1$ means it is false. No independence is assumed: the coordinates of $H^{(n)}$ may be dependent, and the conditional law of $X^{(n)}$ given $H^{(n)}$ may have arbitrary dependence across coordinates. Let $P_n$ denote the joint law of $(X^{(n)},H^{(n)})$.

A multiple-testing rule is a measurable map $\delta_n:\mathcal X^n\to\{0,1\}^n$, with $\delta_{n,i}(X^{(n)})=1$ meaning reject hypothesis $i$. Define

Here $V_n$ is the number of false discoveries and $W_n$ is the number of false non-discoveries. The false discovery rate and false non-discovery rate under $P_n$ are

Fix $\alpha\in(0,1)$. for a dependent model sequence $P=(P_n)_{n\ge1}$, define