Unique median of a sequence with no limit

Question

I have a sequnce of random variables $(X_n$), I know that cumulative distribution functions $F_{X_n}(r) \to F_x(r), n \to \infty, \forall r $. I need an example of $X_n$, such that their medians are unique, but they do not have a limit.

Answer 1

Here is one of solutions; G.Grimmett&D.Stirzaker's "One thousand exercises" provides another one.

Suppose $\varepsilon_n\searrow 0$. For even $n$ let $\mathsf{P}(X_n\in [0,1])=\mathsf{P}(X_n\in [2,3]) = \frac12-\varepsilon_n$ and $\mathsf{P}(X_n\in [1,\frac32])=2\varepsilon_n$, where $X_n$ is uniformly distributed over each interval. Then $med(X_n)=\frac54$. For odd $n$ distribution of $X_n$ is the same with replacement $[1,\frac32]$ by $[\frac32, 2]$. Here $med(X_n)=\frac74$.

One can see that $X_n$ weakly converges, but medians don't have any limit.

Ah, and you failed the course, Kamil.