# Is there a central limit theorem for Median?

1. Apr 19, 2010

### grossgermany

1. The problem statement, all variables and given/known data
Hi, We know the famous central limit theorem for means.
I wonder if there is a central limit theorem for Median?
If so under what regularity condition, does the median converge to a normal distribution with mean and variance equal to what?

2. Relevant equations

3. The attempt at a solution

2. Apr 19, 2010

One of the simplest results is this:
* Assume the model is $F(x - \theta)$
* Assume both $F(0) = 1/2$ and that the density $f(0) > 0$

(these assumptions mean the population median is unique)

Then the sample median satisifies

$$\sqrt n \left(\hat \theta - \theta\right) \rightarrow n(0, \sigma^2)$$

in distribution, where the asymptotic variance is given by

$$\sigma^2 = \frac 1 {4f^2(0)}$$