---
# System prepended metadata

title: Student-t asymptotics

---

# Student-t asymptotics

The Student-t distribution pdf is:
$$
p(x) = \frac{\Gamma({0.5(v + 1))}}{\sqrt{v\pi}\Gamma(0.5v)}(1 + \frac{x^2}{v})^{-\frac{v + 1}{2}} \propto (1 + \frac{x^2}{v})^{-\frac{v + 1}{2}}.
$$

So taking $v = \frac{1}{\lambda}$, asymptotically for large $x$, we have 
$$\begin{align}
p(x) &\approx ... + x^{-v-1} = ... + x^{-\frac{1}{\lambda}-1}.
\end{align}$$

Considering the derivative of our "tail transformation" from uniform base, the derivative of the inverse transformation is:
$$\frac{\partial T^{-1}}{\partial x}(x) = (1 + \lambda x)^{-\frac{1}{\lambda} - 1},$$
We see that our constructed density $q(x)$ has the same asymptotics,
$$\begin{align}
q_x(x) &= q_u(T^{-1}(x))\frac{\partial T^{-1}}{\partial x}(x) \\
&= (1 + \lambda x)^{-\frac{1}{\lambda} - 1} \\
&\approx ... + x^{-\frac{1}{\lambda} - 1}
\end{align}$$

Does this mean that the Student-T can have any tail behaviour?