I try to prove the asymptotic normality from the Frechet differentiability. Consider
$$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$
and
$$L_{F}\left(F_{n}-F\right)=L_{F}\left(\frac{1}{n}\sum_{i=1}^{n}\left(\delta_{x_{i}}-x\right)\right)=\frac{1}{n}\sum_{i=1}^{n}L_{F}(\delta_{x_{i}}-F)=\frac{1}{n}\sum_{i=1}^{n}T_{x_{i}}^{\prime}(F)$$
Then, after the simplification, I get
\begin{eqnarray*} T(F_{n})-T(F) & = & L_{F}(F_{n}-F)+o\left(d_{\star}(F_{n},F)\right),\\ \sqrt{n}\left(T(F_{n})-T(F)\right) & = & \sqrt{n}\left[L_{F}(F_{n}-F)+o\left(d_{\star}(F_{n},F)\right)\right],\\ & = & \sqrt{n}L_{F}(F_{n}-F)+\sqrt{n}o\left(d_{\star}(F_{n},F)\right),\\ & = & \frac{1}{\sqrt{n}}\sum_{i=1}^{n}T_{x_{i}}^{\prime}(F)+\sqrt{n}o\left(d_{\star}(F_{n},F)\right),\\ & = & \frac{1}{\sqrt{n}}\sum_{i=1}^{n}T_{x_{i}}^{\prime}(F)+o_{p}(1). \end{eqnarray*}
However, from the last formula, I do not see the asymptotic normality. Could someone help me, how from the last formula the asymptotic normality can be proven?
Thanks for your help.