For $k\geq 1$ let $T_k\sim \chi^2_k$.
For $a\in ]0,1[$ let us define $c_a(k) = \min \{ s\geq 0, \mathbf P ( T_k \leq s )\geq a\}$.
Would anybody have a free reference for a study of the asymptotics of the sequence $(\ c_a(k)\ )_k$ ?
It looks like for $k\to+\infty$, it should begin with something like
$$c_a(k) = k + p(a) \sqrt k + o(\sqrt k)$$
I would already be happy to see a formula for $p(a)$, but the dominant term in the error $o(\sqrt k)$ is also of interest to me ...
Maybe the solution is written in this paper but it is not free.
Thanks to @Henry. The result is thus :
$$c_a(k)=k+z_a\sqrt{2k}+O(1)$$
where $z_a$ is defined by using $Y\sim\mathcal N(0,1)$ as
$$z_a=\min\{s≥0,\mathbf P(Y≤s)≥a\}$$