Lévy's continuity theorem
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.
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 φ
then the following statements become equivalent:
- converges in distribution to some random variable X
- is tight:
- φ(t) is a characteristic function of some random variable X;
- φ(t) is a continuous function of t;
- φ(t) is continuous at t = 0.
- Williams (1991, section 18.1)
- Fristedt & Gray (1996, Theorem 18.21)
- Williams, D. (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6.
- Fristedt, B.E.; Gray, L. F. (1996): A modern approach to probability theory, Birkhäuser Boston. ISBN 0-8176-3807-5
- Lecture notes of "Theory of Probability" from MIT Open Course Sessions 9–14 are related to this theorem.