Continuous mapping theorem

Remarks

Part 3 is direct: on the event where $X_n \to X$ and $X \notin D_g$, the continuity of $g$ at $X$ gives $g(X_n) \to g(X)$, and this event has probability $1$. Part 1 follows from the portmanteau theorem: for a closed set $F \subseteq S'$, the closure of $g^{-1}(F)$ lies in $g^{-1}(F) \cup D_g$. Part 2 follows from part 3 applied along subsequences, since convergence in probability holds exactly when every subsequence has a further subsequence that converges almost surely.

Example: if $X_n \to Z$ in distribution with $Z$ standard normal, then $X_n^2$ converges in distribution to a chi-squared variable with one degree of freedom.

The hypothesis $P(X \in D_g) = 0$ cannot be dropped. Take $X_n = 1/n$, $X = 0$ and $g = \mathbf{1}_{(0, \infty)}$. Then $X_n \to X$ surely, but $g(X_n) = 1$ for all $n$ while $g(X) = 0$. Here $D_g = \set{0}$ and $P(X \in D_g) = 1$.

Going deeper

For a student

The theorem is the main tool to move limit results from one statistic to another. With Slutsky's theorem and the central limit theorem, it gives the limit laws of sums, products, quotients and squares of asymptotically normal statistics. It also underlies the delta method, where $g$ is a differentiable function. For random paths, applied with $g$ the supremum map, it turns Donsker's theorem into a limit law for the maximum of a random walk.