Lévy's continuity theorem

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In probability theory, Lévy’s continuity 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. An alternative name sometimes used is Lévy’s convergence theorem.[1]

This theorem is the basis for one approach to prove the central limit theorem and it is one of the major theorems concerning characteristic functions.


Suppose we have

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

\varphi_n(t)\to\varphi(t) \quad \forall t\in\mathbb{R},

then the following statements become equivalent:


Rigorous proofs of this theorem are available.[1][2]


  1. ^ a b Williams (1991, section 18.1)
  2. ^ Fristedt & Gray (1996, Theorem 18.21)