test problem described in Section \[toy\_model\], and show this in Figure \[fig:toy\_model\_piecewise\_const\].

![image](Fig1.eps){width="\textwidth"}

Assume $S \subset \mathbb{R}^s$ is compact and that a unique $\mathbf{w}^* \in E$ maximises $D_h\left(p || q^{(\mathbf{w})} \right)$. Assume the conditions of Lemma 1 hold for distributions $p$ and $q^{(\mathbf{w})}$. Suppose $L(\mathbf{w})$ is piecewise constant in $\mathbf{w}$ with finitely many jump discontinuities. Then $$\lim_{n \rightarrow \infty} \mathbb{P} \left( \operatorname*{\arg\!\max}_{\mathbf{w} \in E} \hat{D}_h \left( X_{1:n} || Y_{1:n}^{(\mathbf{w})} \right) = \mathbf{w}^* \right) = 1.$$

We can choose a $\delta >0$ and finitely many $\mathbf{w}^j$, $j \in \{1, \dots, \chi \}$ such that every point $\mathbf{w}$ is within a ball of radius $\delta$ from some $\mathbf{w}^j$ (see Figure \[fig:toy\_model\_piecewise\_const\] and note that $L$ is locally constant). Then $\forall \mathbf{w} \in E \, \, ||\mathbf{w}-\mathbf{w}^j|| < \delta \, \implies \, L(\mathbf{w}) = L(\mathbf{w}^j) = L_j. $ Since there are finitely many values $\mathbf{w}^j$ corresponding to distinct unique values $L^j$, we can apply the result from Theorem 3 to give the required result.

Suppose that $k\ge 2$ and that $\mathcal{M} = \text{supp}(p)$. Assume that (a) $q$ is bounded above, (b) $p$ is bounded away from zero, (c) $p$ is uniformly Lebesgue approximable, (d) $\exists \, \delta_0$ such that $\forall \delta \in (0,\delta_0) 