Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function

Question

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 $F_{t_u}^{-1}(F_{t_v}(x)) =\mathcal{O}(x^k\exp{(x)})$)

Answer 1

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 $|x|\to\infty$.