Timeline for Limit of $t$-distribution as $n$ goes to infinity
Current License: CC BY-SA 4.0
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| May 20, 2021 at 2:01 | comment | added | whuber♦ | @Danny Since $$\log f_n(t)=-\frac{n+1}{2}\log(1+t^2/n)$$plus other stuff and the expression shown equals $-(n+1)t^2/(2n)+o(1/n),$ which converges to $-t^2/2,$ $f_n(t)$ must converge to some multiple of $\exp(-t^2/2).$ The multiple $\sqrt{1/(2\pi)}$ is determined by the normalization to unity. Thus, you never have to compute the limit of the Gamma functions -- indeed, you can now infer from this result that $$\lim_{n\to\infty} \frac{\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{\pi n}\,\Gamma\left(\frac{n}{2}\right)}=\frac{1}{\sqrt{2\pi}}.$$ | |
| Apr 8, 2021 at 0:07 | comment | added | Danny | Very nice answer. Is there any way to get at the convergence of the density functions without considering the densities explicitly? I can't seem to come up with anything constructive. | |
| Jun 11, 2020 at 14:32 | history | edited | CommunityBot |
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| Feb 12, 2020 at 15:00 | history | answered | whuber♦ | CC BY-SA 4.0 |