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.[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.
Theorem
Suppose we have
- a sequence of random variables , not necessarily sharing a common probability space,
- the sequence of corresponding characteristic functions , which by definition are
If the sequence of characteristic functions converges pointwise to some function
then the following statements become equivalent:
- converges in distribution to some random variable X
- is tight:
- is a characteristic function of some random variable X;
- is a continuous function of t;
- is continuous at t = 0.
Proof
Rigorous proofs of this theorem are available.[1][2]
Notes
References
- 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
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