Fix $q\in (0,1)$, $k\in\mathbb{N}_{\ge 1}$ and let $F_k$ and $\Phi$ denote the cdf's of the $\chi_k^2$ and standard normal distribution, respectively (and $F_k^{-1}$ the inverse function of $F_k$). I'm trying to show that
$$\lim_{k\to\infty} \frac{F_k^{-1}(q)-k}{\sqrt{2k}}=\Phi^{-1}(q).$$
Since $E(\chi_k^2)=k$ and $Var(\chi_k^2)=2k$ this of course looks like an application of the CLT but as far as I can see we can only apply the CLT to the cdf, not its inverse.
Any hints on how to continue are appreciated.
By the CLT, $F_k((x-k)/\sqrt{2k})\to\Phi(x)$ as $k\to\infty$. Wouldn't this mean that $k+\sqrt{2k}F_k^{-1}(y)$ is approximately $\Phi^{-1}(y)$ as $k\to\infty$?