Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables such that almost surely $X_n \rightarrow X$.
Given just the information above (i.e. no information about distribution) can one determine the median of $X_n$ in the limit as $n \rightarrow \infty$? Or could anyone give me an appropriate reference?
The median $M_n$ converges almost surely to $X$. Proof: for every $x\gt0$, $X_n\leqslant X-x$ for at most finitely many $n$ hence $M_n\geqslant X-x$ for every $n$ large enough. Likewise, for every $x\gt0$, $X_n\geqslant X+x$ for at most finitely many $n$ hence $M_n\leqslant X+x$ for every $n$ large enough.
This uses only that $X_n\to X$ almost surely. Note that this assumption does not hold if $(X_n)$ is i.i.d. (except for almost surely constant random variables) hence the statement of the problem is inconsistent.
