Information-theoretic limits under heavy-tailed (non-sub-Gaussian) noise

Fix integers $d\ge 1$ and $N\ge 1$, and fix contamination parameters $\kappa\in(0,1/2]$ and $\epsilon\in[0,1/2-\kappa]$ (equivalently, one may consider the boundary regime $\epsilon\le 1/2$). Let $p>2$, $\nu>0$, and let $K\subseteq\mathbb R^d$ be a known constraint set for the unknown mean vector $\mu$ (in the motivating framework, $K$ is star-shaped with respect to $0$). Define

For $(\mu,P)\in K\times\mathcal P_{p,\nu}$, let $X_i^\star=\mu+\xi_i$ with $\xi_1,\dots,\xi_N\stackrel{i.i.d.}{\sim}P$. Under Huber contamination, an adversary replaces at most $\lfloor\epsilon N\rfloor$ observations arbitrarily, yielding $X_1,\dots,X_N$. For measurable estimators $\hat\mu:(\mathbb R^d)^N\to\mathbb R^d$ (or $K$), consider