# Using Student Distribution to obtain the Gaussian Distribution

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?

• Commented May 13, 2020 at 4:23
• Commented May 13, 2020 at 4:32

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.
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.