\, \int_{\mathcal{M}} H(x,p,\delta,1/2)p(x) \mathrm{d}x < \infty,$ (e) $\int_{\mathcal{M}}||x-y||^{\gamma} p(y) \mathrm{d}y < \infty$ for almost all $x \in \mathcal{M}$, $\int \int_{\mathcal{M}^2}||x-y||^{\gamma} p(y) p(x) \mathrm{d}y \mathrm{d} x < \infty$. Assume $S \subset \mathbb{R}^s$ is compact and that a unique $\mathbf{w}^* \in E$ maximises $D_h\left(p || q^{(\mathbf{w})} \right)$.

Then $$\lim_{n \rightarrow \infty} \mathbb{P} \left( \operatorname*{\arg\!\max}_{w \in E} \hat{D}_h \left( X_{1:n} || Y_{1:n}^{(\mathbf{w})} \right) = \mathbf{w}^* \right) = 1$$

Apply the Lemma 4 to show that $L(\mathbf{w})$ is piecewise constant in $\mathbf{w}$. Apply Lemma 5 to give the desired result.

To summarise, with this nearest neighbour estimator, under conditions on $p$ and $q$, we have $L_2$ convergence of the estimator and this ensures that, in the limit of more samples, estimates of the distance between $p$ and $q$ will become more concentrated around the true distance, such that optimising the estimate of the distance will give the true optimum, $\mathbf{w}^*$, by making use of the piecewise constant structure of $L(\mathbf{w})$. We assume in Theorem 2 (and elsewhere) that the space of parameters, $S$, is compact. In practice this is not a problem, since we can work with a constrained optimization problem and assume that the summary statistic weights lie within a large but finite 