Using Student Distribution to obtain the Gaussian Distribution

Question

From the Student distribution:

$$\rho(x) = \frac{1}{\sqrt{N\pi}} \frac{\Gamma\big(\frac{N+1}{2} \big)}{\Gamma\big(\frac{N}{2} \big)} \bigg(1+ \frac{x^{2}}{N} \bigg)^{-(N+1)/2} $$

it's possible to obtain the Gaussian distribution by doing $N \rightarrow \infty $.

I did the limit by separating it in two functions:

$$ \lim_{N \rightarrow \infty} g(N)f(N) = \lim_{N \rightarrow \infty} g(N)\lim_{N \rightarrow \infty} f(N) $$

where:

$$ g(N) = \frac{1}{\sqrt{N\pi}} \frac{\Gamma\big(\frac{N+1}{2} \big)}{\Gamma\big(\frac{N}{2} \big)} $$

and

$$ f(N) = \bigg(1+ \frac{x^{2}}{N} \bigg)^{-(N+1)/2} $$

I know they exist, so this action holds. The $f(N)$ limit is equal to:

$$ \lim_{N \rightarrow \infty} \bigg(1+ \frac{x^{2}}{N} \bigg)^{-(N+1)/2} = e^{-x^{2}/2} $$

but, the $g(N)$ limit is where I get stuck on:

$$ \lim_{N \rightarrow \infty} \frac{1}{\sqrt{N\pi}} \frac{\Gamma\big(\frac{N+1}{2} \big)}{\Gamma\big(\frac{N}{2} \big)} $$

This limit is one of the type $\frac{\infty}{\infty}$. I know that the gamma function has derivatives involving the polygamma function, like:

$$ \Gamma'(z) = \Gamma(z)\psi^{(0)}(z) $$

where,

$$ \psi^{(m)}(z) = \frac{d^{m+1}}{dz^{m+1}} \ln \Gamma(z) $$

or, more generally,

$$ \frac{d^{m}}{dz^{m}} \Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t}(\ln t)^{m} dt,$$

where, $R(z) > 0$.

So, there are anyway of going forward?

Or maybe, anyone knows a better path?

Answer 1

@StubbornAtom gave you the proper links.

When you face ratio's of gamma functions or factorials, the trick is to take the logarithms and use Stirling approximation of $\log(\Gamma(p))$. Apply it (as many times as required) replacing $p$ by the proper argument, continue with Taylor series and exponentiate it using Taylor series again. I shall let you doing it since it is a good exercise you will practice very often.

If I may add a comment, what you did separating the functions to get the limit is very correct but you just end with the limit (!!).

You could have obtained a bit more of information working with the whole problem. $$\rho(x) = \frac{1}{\sqrt{N\pi}} \frac{\Gamma\big(\frac{N+1}{2} \big)}{\Gamma\big(\frac{N}{2} \big)} \bigg(1+ \frac{x^{2}}{N} \bigg)^{-(N+1)/2}$$ $$\log(\rho(x))=-\frac 12 \log(N)-\frac 12 \log(\pi)+\log\left(\Gamma\big(\frac{N+1}{2} \big)\right)-\log\left(\Gamma\big(\frac{N}{2} \big)\right)-\frac{N+1}2 \log\left(1+ \frac{x^{2}}{N}\right)$$ Do what I did suggest and you will end with $$\rho(x)=\frac{e^{-\frac{x^2}{2}}}{\sqrt{2 \pi }}\left(1+\frac{x^4-2 x^2-1}{4 N}+O\left(\frac{1}{N^2}\right)\right)$$ which gives an idea about the deviation from the gaussian.