Lévy's continuity theorem explained

In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem,^[1] 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

If the sequence of characteristic functions converges pointwise to some function

\varphi \varphi _{n(t)\to\varphi(t)} \forall t\in R , then the following statements become equivalent:

Proof

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

Notes and References

1. Book: Williams, D. . David Williams (mathematician)

   . David Williams (mathematician) . Probability with Martingales . 1991 . Cambridge University Press . 0-521-40605-6 . section 18.1 .

2. Book: Fristedt, B. E. . Gray . L. F. . 1996 . A modern approach to probability theory . Birkhäuser . Boston . 0-8176-3807-5 . Theorems 14.15 and 18.21 .