Let's denote


* $F\_{t\_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t\_u$ with $u$ degrees of freedom and
* $F\_{t\_v}(x)$ the cumulative distribution function of the t-distribution $t\_v$ with $v$ degrees of freedom


where $u \ne v$ and $u,v >2$.


What is the asymptotic behavior of the function $F\_{t\_u}^{-1}(F\_{t\_v}(x))$ when $x \rightarrow \pm \infty$ ? ( just for example $F\_{t\_u}^{-1}(F\_{t\_v}(x)) =\mathcal{O}(x^k\exp{(x)})$)



Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function
Let $F\_u:=F\_{t\_u}$ and $G\_u:=1-F\_u$. Let $f\_u:=F'\_u$, the pdf of $t\_u$, so that
$$f\_u(x)=c\_u(1+x^2/u)^{-(u+1)/2},\quad c\_u:=\frac{\Gamma((u+1)/2)}{\Gamma(u/2)\sqrt{\pi u}}.$$
So, for $x\to\infty$
$$f\_u(x)\sim c\_u\,u^{(u+1)/2}x^{-u-1},$$
whence
$$G\_u(x)=\int\_x^\infty f\_u(y)\,dy
\sim c\_u\,u^{(u+1)/2}\int\_x^\infty y^{-u-1}\,dy
=b\_u\,x^{-u},$$
where
$$b\_u:=c\_u\,u^{(u-1)/2}.$$


The equation $z=F\_u^{-1}(F\_v(x))$ can be rewritten as $F\_v(x)=F\_u(z)$ and then as $G\_v(x)=G\_u(z)$, whence for $x\to\infty$
$$b\_u\,z^{-u}\sim b\_v\,x^{-v},$$
so that
$$F\_u^{-1}(F\_v(x))=z\sim (b\_v/b\_u)^{1/u}\,x^{v/u}.$$
So, by symmetry,
$$F\_u^{-1}(F\_v(x))\sim (b\_v/b\_u)^{1/u}\,|x|^{v/u}$$
as $|x|\to\infty$.


