1) $f_{X}$, $f_{Y}$, $f_{XY}$, and $g$ are smooth; 2) $f_{X}$ and $f_{Y}$ have bounded support sets $\mathcal{S}_{X}$ and $\mathcal{S}_{Y}$; 3) $f_{X}$, $f_{Y}$, and $f_{XY}$ are strictly lower bounded on their support sets. More specifically, we assume that the densities belong to the bounded Hölder class $\Sigma(s,H)$ (the precise definition is included in the appendices) which implies that the densities are $r=\left\lfloor s\right\rfloor $ times differentiable. These assumptions are comparable to those in similar studies on asymptotic convergence analysis [@moon2016isit; @moon2014nips; @moon2014isit; @singh2014renyi; @singh2014exponential; @sricharan2013ensemble; @krishnamurthy2014divergence; @kandasamy2015nonparametric]. To derive the convergence rates without boundary corrections, we also assume that 4) the boundary of the support set is smooth with respect to the corresponding kernels. The full assumptions are

-   $(\mathcal{A}.0)$: The kernels $K_{X}$ and $K_{Y}$ are symmetric product kernels with bounded support.

-   $(\mathcal{A}.1)$: There exist constants $\epsilon_{0},\epsilon_{\infty}$ such that $0<\epsilon_{0}\leq f_{X}(x)\leq\epsilon_{\infty}<\infty,\,\forall x\in\mathcal{S}_{X}$, $\epsilon_{0}\leq f_{Y}(y)\leq\epsilon_{\infty},\,\forall y\in\mathcal{S}_{Y}$, and $\epsilon_{0}\leq f_{XY}(x,y)\leq\epsilon_{\infty},\,\forall(x,y)\in\mathcal{S}_{X}\times\mathcal{S}_{Y}$.

-   $(\mathcal{A}.2)$: Each of the densities belong to $\Sigma(s,H)$ in the interior of their support sets with $s\geq2$.

-   $(\mathcal{A}.3)$: $g\left(t_{1}/t_{2}\right)$ has an infinite number of mixed derivatives wrt $t_{1}$ and $t_{2}$.

-   $(\mathcal{A}.4$): $\left|\frac{\partial^{k+l}g(t_{1},t_{2})}{\partial t_{1}^{k}\partial t_{2}^{l}}\right|/(k!l!)$, $k,l=0,1,\ldots$ are strictly upper bounded for $\epsilon_{0}\leq t_{1},t_{2}\leq\epsilon_{\infty}$.

-   $(\mathcal{A}.5)$: Let $K$ be either $K_{X}$ 