# Math Help - Function of quantiles converges in law to a normally distributed random variable

1. ## Function of quantiles converges in law to a normally distributed random variable

The challenge here is to show that

$\left(\frac{n}{p(1-p)}\right)^{\frac{1}{2}}f\left(T_p\right)\left(\ch i _{(p)}-T_p\right)\to Z\sim N(0,1)$

Where convergence here is convergence in law, T_p is the pth quantile of f, where f is the PDF of the F distribution and r/n=p remains fixed. I don't even know where to start! Many thanks to whomever can solve this toughie...

2. Use the fact that if $\chi_{(p)}$ is the sample quantile (sample size n) and $T_p$ is the population quantile (both are pth quantile) then asymptotically $\chi_{(p)}\sim N(T_p,\frac{p(1-p)}{n[f(\chi_{(p)})]^2})$. You have to standardize it.