Let $X$,$Y$ two real valued random variables with distributions absolutely continuous with regard to the Lebesgue measure. Denote their distribution functions $F$ and $G$, and $g=G'$ the probability density function of $Y$. Using the inverse transform theorem, we have:
\begin{align*} Y &= G^{-1}\circ F (X)\\ & =: f(X). \end{align*} in distribution. Now, one observes $(X_n)_{n\geq1}$ i.i.d. $X$ and $(Y_n)_{n\geq1}$ i.i.d. Y. Suppose moreover that $(X_n)_{n\geq1}$ is independant of $(Y_n)_{n\geq1}$. Define $\hat{f}_n(x)$ an estimator of $f(x) = G^{-1}\circ F(x)$ for some $x\in \mathbb{R}$ as: \begin{align*} \hat{f}_n(x) = G_n^{-1}\circ F_n(x) \end{align*} where $F_n(x)=n^{-1}\sum_{i=1}^n \boldsymbol{1}_{\{X_i\leq x\}}$ is the empirical c.d.f. of the $n$-sample $(X_1, \dots, X_n)$ and $G_n^{-1}$ is the generalized inverse of the c.d.f. of the $n$-sample $(Y_1,\dots,Y_n)$, i.e. $G_n^{-1}(\alpha) = \inf\{x \in \mathbb{R}, G_n(x) \geq \alpha \}$ where $G_n(x)=n^{-1}\sum_{i=1}^n \boldsymbol{1}_{\{Y_i\leq x\}}$.
I am interested in the asymptotic behavior of $\hat{f}_n(x)$. One can write: \begin{align*} \sqrt{n}(\hat{f}_n(x) - f(x)) &= \sqrt{n}(G_n^{-1}\circ F_n(x) - G^{-1}\circ F(x))\\ &= \sqrt{n}(G_n^{-1}\circ F_n(x) - G^{-1}\circ F_n(x))+\sqrt{n}(G^{-1}\circ F_n(x) - G^{-1}\circ F(x))\\ &= A_n + B_n \quad \text{say.} \end{align*} Proving stochastic equicontinuity of the process $\{Z_n(p), p \in [0,1]\}_{n \geq 1}$ where $Z_n(p) = \sqrt{n}(G_n^{-1}(p) - G^{-1}(p))$ allows one to prove that $A_n$ converges in distribution to $\mathcal{N}(0,\sigma^2)$ where $\sigma^2 = \frac{F(x)(1-F(x))}{g\circ f(x)^2}$. Moreover the Delta method can be used to prove that $B_n$ converges in distribution also to $\mathcal{N}(0,\sigma^2)$.
Now, note that that $A_n$ and $B_n$ are not independent so we can not conclude that $A_n + B_n$ converges to $\mathcal{N}(0,2\sigma^2)$. Nevertheless, numerical simulations are consistent with this.
Any idea on how to prove it would be greatly appreciated! Thanks!