Continuity Theorem

Last updated Nov 1, 2022

Let $(X_{n}){n=1}^{\infty}$ be a Sequence of Random Variable (on arbitrary Probability Spaces) so that $X{n}$ has Characteristic Function $\phi_{n}$ for all $n \in \mathbb{N}$.

1. If $X_{\infty}$ is a Random Variable with Characteristic Function $\phi_{\infty}$ and $X_{n} \Rightarrow X_{\infty}$, then $$\phi_{n} \to \phi_{\infty}$$ pointwise on $\mathbb{R}$.
2. Suppose $\phi_\infty(t) := \lim\limits_{n \to \infty} \phi_{n}(t)$ exists for all $t \in \mathbb{R}$ and $\phi_\infty$ is continuous at $0$. Then $\phi_{\infty}$ is the Characteristic Function of a Random Variable $X_{\infty}$ and $X_{n} \Rightarrow X_{\infty}$.

TODO - See thm 9.5.2 in Resnick - A Probability Path page 304