Lévy's continuity theorem

Template:Short description In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. This theorem is the basis for one approach to prove the central limit theorem and is one of the major theorems concerning characteristic functions.

Statement

Suppose we have Template:Unordered list

If the sequence of characteristic functions converges pointwise to some function $\phi$

${\phi }_n(t)\to \phi (t)\mathrm{\forall }t\in \mathbb{ℝ},$

then the following statements become equivalent: Template:Unordered list

Proof

Rigorous proofs of this theorem are available.