Lévy's continuity theorem

Statement

Let $X, X_1, X_2, \dots$ be real random variables with characteristic functions $\varphi, \varphi_1, \varphi_2, \dots$.

1. If $X_n \to X$ in distribution, then $\varphi_n(t) \to \varphi(t)$ for each $t \in \mathbb{R}$.
2. Conversely, suppose that $\varphi_n(t)$ converges to a limit $\psi(t)$ for each $t \in \mathbb{R}$, and that $\psi$ is continuous at $0$. Then $\psi$ is the characteristic function of a random variable $Y$, and $X_n \to Y$ in distribution.

Remarks

Part 1 follows from the definition of convergence in distribution with the bounded continuous functions $\cos(tx)$ and $\sin(tx)$. The continuity at $0$ in part 2 is necessary. For $X_n \sim \mathcal{N}(0, n)$, $\varphi_n(t) = e^{-nt^2/2}$ tends to $1_{\{t = 0\}}$, which is not continuous at $0$, and $X_n$ does not converge in distribution. The theorem holds in the same form for random vectors in $\mathbb{R}^k$.

History

The theorem is named after Paul Lévy.

Explanations

L1 human

To show that random variables converge in distribution, it is enough to show that their characteristic functions converge. This is often much easier, because characteristic functions of sums of independent variables are products.