Continuous Mapping Theorem

Last updated Nov 1, 2022

Let $({X_n}{n}){n=0}^{\infty}$ be a Sequence of Random Variables such that $$X_{n} \Rightarrow X_{0}.$$ Assume $F_{n}$ is the Probability Distribution Function for $X_{n}$ for $n \geq 0$. Let $h: \mathbb{R} \to \mathbb{R}$ be s.t. $$\mathbb{P}[X_{0} \in \text{Disc}(h)] = 0.$$ Then $$h(X_{n}) \Rightarrow h(X_{0}).$$ If $h$ is bounded, then $$\mathbb{E}(h(X_{n})) = \int\limits h(x) F_{n}(dx) \to \mathbb{E}(h(X_{0})) = \int\limits h(x) F_{0}(dx)$$

TODO

• Baby Skorohod Theorem

See Resnick - A Probability Path Cor 8.3.1 pg 261