Median of large numbers of gaussian has gaussian distribution Prove that for large n, median of n independent Gaussian data with mean $\mu$ and variance $ \sigma ^ 2 $ has approximately normal distribution with mean $\mu$ and variance $ ((\pi * (\sigma)^2 ) / (2*n) ) $ . What about non-gaussian data?
This is a homework. I think I should use the central law theorem or law of large numbers. 
 A: Slightly abridged from Bain and Englehardt: Intro to probability theory and mathematical statistics, Sect 7.4. (Asymptotic Distribution of Central Order Statistics.)
Let $X_1, \dots, X_n$ be a random sample from a continuous PDf $f(x)$ that
is continuous and nonzero at the $p$th percentile $x_p,$ for $0 < p < 1.$
If $k/n \rightarrow p$ (with $k =np$ bounded), then the sequence of the $k$th order statistics $X_{k:n}$ is asymptotically normal with mean $x_p$ and
variance $c^2/n,$ where $c^2 = \frac{p(1-p)}{[f(x_p)]^2}.$ (A proof for a
specific non-normal case is given.) Notice that the theorem does not apply to the maximum or minimum. 
The version for the sample median is sometimes called the CLT for the sample median.
The condition for the usual CLT for sample means is that $Var(X_i)$ is finite.
This corresponding theorem for sample medians requires that the PDF is not
$0$ in the vicinity of the population median $\eta,$ so that a unique population median exists.
For example, if $X_i \stackrel{iid}{\sim} \mathsf{Exp}(1),$ then the sample median is asymptotically normal with mean $\ln 2$ and asymptotic variance $1/n.$
The following simulation in R illustrates that $n = 100$ is not quite a large
enough sample size for a really good fit of sample medians to $\mathsf{Norm}(\mu=\log 2,\, \sigma=1/10).$ (The distribution of sample medians $H$ is slightly right-skewed.)
m = 10^5; n = 100
h = replicate(m, median(rexp(n)))
mean(h);  sd(h)
## 0.6982754  # compare ln 2 = 0.693
## 0.09977901 # compare .1
hist(h, prob=T, col="skyblue2", main="Sample Medians of Sample of 100 from EXP(1)")
curve(dnorm(x, log(2), sqrt(1/n)), add=T, lwd=2)
abline(v=log(2), lwd=2, col="red", lty="dotted")


