Let $X_1,X_2,\ldots$ be a random sample from the uniform distribution on the interval $(0,1)$. Assuming that $n$ is odd, find the pdf of the sample median (say $M_n$). Does the pdf of the r.v. $(M_n-EM_n)/\sqrt{var(M_n)}$ converge to a limit as $n\rightarrow\infty$?
My work: I find $M_n\sim Beta(\frac{n+1}{2},\frac{n+1}{2})$, $E M_n=1/2$, and $var(M_n)=\frac{1}{4(n+2)}$. I do not know use which theorem to solve the limit problem.
The PDF of $M_n$ is $$ \frac1{\beta(\tfrac{n+1}2,\tfrac{n+1}2)} x_+^{(n-1)/2} (1-x)_+^{(n-1)/2} $$ where $x_+$ is defined to be $x$ if $x > 0$, and zero otherwise.
So the PDF of $(M_n - EM_n)/\sqrt{\text{var}(M_n)}$ is $$ \frac{1}{\sqrt{4(n+2)} \beta(\tfrac{n+1}2,\tfrac{n+1}2)} \left(\frac12 + \frac x{\sqrt{4(n+2)}}\right)_+^{(n-1)/2} \left(\frac12 - \frac x{\sqrt{4(n+2)}}\right)_+^{(n-1)/2} \\ = \frac1{2^{n-1}\sqrt{4(n+2)} \beta(\tfrac{n+1}2,\tfrac{n+1}2)} \left(1 - \frac {x^2}{n+2}\right)_+^{(n-1)/2} $$ As $n\to \infty$, and using Stirling's Formula, we see that this converges to $$ \frac1{\sqrt{2\pi}} e^{-x^2/2} .$$