Lévy's continuity theorem
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Short description: Result in probability theory
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
- a sequence of random variables [math]\displaystyle{ \{X_n\}_{n=1}^\infty }[/math], not necessarily sharing a common probability space,
- the sequence of corresponding characteristic functions [math]\displaystyle{ \{\varphi_n\}_{n=1}^\infty }[/math], which by definition are
- [math]\displaystyle{ \varphi_n(t) = \operatorname{E} \left[ e^{itX_n} \right] \quad \forall t\in\mathbb{R},\ \forall n\in\mathbb{N}, }[/math]
If the sequence of characteristic functions converges pointwise to some function [math]\displaystyle{ \varphi }[/math]
- [math]\displaystyle{ \varphi_n(t)\to\varphi(t) \quad \forall t\in\mathbb{R}, }[/math]
then the following statements become equivalent:
- [math]\displaystyle{ X_n }[/math] converges in distribution to some random variable X
- [math]\displaystyle{ X_n\ \xrightarrow{\mathcal D}\ X, }[/math]
- [math]\displaystyle{ \{X_n\}_{n=1}^\infty }[/math] is tight:
- [math]\displaystyle{ \lim_{x\to\infty}\left( \sup_n \operatorname{P}\big[\, |X_n|\gt x \,\big]\right) = 0; }[/math]
- [math]\displaystyle{ \varphi(t) }[/math] is a characteristic function of some random variable X;
- [math]\displaystyle{ \varphi(t) }[/math] is a continuous function of t;
- [math]\displaystyle{ \varphi(t) }[/math] is continuous at t = 0.
Proof
Rigorous proofs of this theorem are available.[1][2]
References
- ↑ 1.0 1.1 Probability with Martingales. Cambridge University Press. 1991. section 18.1. ISBN 0-521-40605-6.
- ↑ Fristedt, B. E.; Gray, L. F. (1996). A modern approach to probability theory. Boston: Birkhäuser. Theorems 14.15 and 18.21. ISBN 0-8176-3807-5.
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