- Lévy continuity theorem
The Lévy continuity theorem in
probability theory , named after the French mathematician Paul Lévy, is the basis for one approach to prove thecentral limit theorem and it is one of the central theorems concerning characteristic functions.Suppose we have
* a sequence ofrandom variable s scriptstyle (X_n)_{n=1}^infty not necessarily sharing a commonprobability space , and
* the corresponding sequence of characteristic functions scriptstyle (varphi_n)_{n=1}^infty, which by definition are
*: varphi_n(t)=E!left(e^{itX_n} ight)quadforall tinmathbb{R}, quadforall ninmathbb{N}(where E is theexpected value operator).The theorem states that if the sequence of characteristic functions converge pointwise to a function scriptstyle varphi , i.e.:forall tinmathbb{R} : varphi_n(t) ovarphi(t)then the following statements become equivalent,* scriptstyle X_n converges in distribution to some
random variable scriptstyle X :X_n xrightarrow{mathcal D} X i.e. the cumulative distribution functions corresponding to random variables converge(see convergence in distribution)* scriptstyle (X_n)_{n=1}^infty is tight, i.e. :lim_{x oinfty}left( sup_n P( |X_n|>x ) ight) = 0
* scriptstyle varphi(t) is a characteristic function of some random variable scriptstyle X.
* scriptstyle varphi(t) is a
continuous function of scriptstyle t.* scriptstyle varphi(t) is
continuous at scriptstyle t=0.An immediate corollary that is useful in proving the
central limit theorem is that, scriptstyle (X_n)_{n=1}^infty converges in distribution to some random variable scriptstyle X with the characteristic function scriptstyle varphi if it is the pointwise convergent limit of scriptstyle (varphi_n)_{n=1}^infty and scriptstyle varphi(t) is continuous at scriptstyle t=0.Proof
Rigorous proof of this theorem is available in "A modern approach to probability theory" by Bert Fristedt and Lawrence Gray (1997): Theorem 18.21
External links
* [http://ocw.mit.edu/OcwWeb/Mathematics/18-175Spring-2007/LectureNotes/index.htm Lecture Note of "Theory of Probability" from MIT Open Course ] Session 9-14 is related to this theorem.
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