Prove that t-distribution is close to normal distribution when degree of freedom is large.
$$f(x)=\frac{\Gamma[(n+1)/2]}{\sqrt{n\pi}\Gamma(n/2)}\left(1+\frac{x^2}{n}\right)^{-(n+1)/2}$$
using limit of density and Law of large number.
Anyone can help me with this question please? Thank you and appreciate your help!
Let us inspect the limit when $df \to \infty$, $$ \lim_{n \to \infty} f(x)=\lim_{n \to \infty}\frac{\Gamma[(n+1)/2]}{\sqrt{n\pi}\Gamma(n/2)}\lim_{n \to \infty}\left(1+\frac{x^2}{n}\right)^{-(n+1)/2}, $$ for the first part you can use Stirling's approximation to note that $$ \lim_{n \to \infty}\frac{\Gamma[(n+1)/2]}{\sqrt{n\pi}\Gamma(n/2)} = \frac{1}{\pi ^{1/2}}\lim_{n \to \infty}\frac{(n/2)^{1/2}}{n^{1/2}} = \frac{1}{\sqrt{2\pi}}, $$ and for the second part you should recall Euler's constant, $$ \lim_{n \to \infty}\left(1+\frac{x^2}{n}\right)^{-(n+1)/2} = e^{-x^2/2}. $$ Thus, $$ \lim_{n \to \infty} f(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}, \quad x\in \mathbb{R} $$
