Example of sample median of continuous random variable that does not converge almost surely I am a bit stuck by the following:
The sample mean generally converges almost surely to the true value given some mild conditions on the distribution;
However, the sample median does not seem to converge almost surely but only in probability. This kind of amazes me, so I would like to know if anyone can give me an example of a continuous random variable $X$ which is normal in a sense such that let's say
$$-\infty < c_1<q_X(0.5-\epsilon)<q_X(0.5+\epsilon)=c_2<\infty$$
so the sample median is somewhat bounded in a finite interval but the sample median does still not converge almost surely..
 A: My guess is that you may be looking for something along the lines of the density $f(x)=\frac32x^2$ on $[-1,1]$.
Since $f(0)=0$, you cannot use Laplace's normal approximation. It seems that for odd sample sizes you get a bimodal distribution for the sample median.
For example, trying in R to look at a sample size of $1001$ with just over $10^5$ simulations , you get the following distribution for the median with a high probability of being  well away from $0$ (and larger sample sizes would only narrow the distribution slowly):
df <- function(x){ ifelse(x < -1,0, ifelse(x > 1,0, 
                   (3/2)*x^2 )) }
pf <- function(q){ ifelse(q < -1,0, ifelse(q > 1,1, 
                   (q^3+1)/2 )) }
qf <- function(p){ ifelse(p < 0,NA, ifelse(p > 1,NA, 
                   sign(2*p-1)*abs(2*p-1)^(1/3) )) }
rf <- function(n){ qf(runif(n)) }
med <- function(n){ median(rf(n)) }
set.seed(2021)
sims <- replicate(100001, med(1001))
summary(sims)
#       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
# -0.5256199 -0.2777905  0.0377480  0.0004548  0.2777602  0.5325429    
plot(density(sims, bw=0.01), main="Simulated distribution of sample median")


