I am trying to find the asymptotic distribution of the estimate of the Cauchy location parameter $\theta$:
$X_\text{least integer $\gt{n\over 2}$}$
I know that for a distribution function $F$, the empirical quantile function has asymptotic distribution
$\sqrt{n}\ (X_\text{least integer $\gt{np}$}-x_p) \overset{d}\longrightarrow \text{N}\left(0,\frac{p(1-p)}{f(x_p)^2}\right)$.
Where $x_p$ is the $p$'th quantile.
My question is, if I can just move around and get the result: $X_\text{least integer $\gt{np}$}\overset{d}\longrightarrow \text{N}(x_p,\frac{p(1-p)}{f(x_p)^2n}+x_p)$.
and then use the $p=1/2$ to get the right result. $X_\text{least integer $\gt{n\over 2}$}\overset{d}\longrightarrow \text{N}(x_{1/2},\frac{1/2(1-1/2)}{f(x_{1/2})^2n}+x_{1/2})=\text{N}(\theta,\frac{1/4}{f(x_{1/2})^2n}+\theta)$.
I am doubting whether this is correct, but I am struggeling to improve it.