Convergence of Random Variables and Median.

Question

Let $\{X_{n}\}_{n}$ be a sequence of real random variables and let $X$ be a real random variable such that $\lim_{n} X_{n}(\omega) = X(\omega)$. Assume that $X_{n}(\omega)$ is a monotone decreasing sequence. My question is whther or not we have that \begin{equation}\nonumber \lim_{n} Med(X_{n}) = Med(X), \end{equation} where $Med(\cdot)$ denotes the univariate median.

Answer 1

It is true. This hypothesis are equivalent to say that $X_{n}$ converges almost surely to $X$. It implies that $X_{n}$ converges in probability to $X$. By Convergence in probability implies convergence of the median? we have that it implies the convergence of the medians.