Lévy's Continuity Theorem

Statement
Consider sequence of random variables $\{X_{n}\}_{n=1}^{\infty}$ with   characteristic functions $\phi_{n}(t):=E[e^{itX_{n}}].$

Then if the characteristic functions converge pointwise to some function $\phi$ i.e. $\displaystyle \phi_{n}(t)\to \phi(t), \forall t\in \mathbb{R}$,

the following statements are equivalent:


 * The $X_{n}$ converge in distribution to some random variable $X$ i.e. $X_{n}\stackrel{dist}{\to} X$ with characteristic function $\phi_{X}(t):=\phi(t)$.


 * The sequence $X_{n}$ is tight i.e. $ \lim\limits_{M\to \infty}\sup_{n\geq 1}P[|X_{n}|\geq M]=0$.