|
|
| Line 1: |
Line 1: |
| In [[probability theory]], '''Lévy’s continuity theorem''', named after the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]], connects [[convergence in distribution]] of the sequence of random variables with [[pointwise convergence]] of their [[Characteristic function (probability theory)|characteristic functions]]. An alternative name sometimes used is '''Lévy’s convergence theorem'''.<ref name=W>Williams (1991, section 18.1)</ref>
| | Hello and welcome. My name is Figures Wunder. Since she was 18 she's been working as a receptionist but her promotion by no means comes. South Dakota is exactly where me and my husband reside. The preferred hobby for my children and me is to play baseball and I'm trying to make it a profession.<br><br>My web-site healthy food delivery - [http://stig.pw/diettogo30380 have a peek at this web-site], |
| | |
| 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
| |
| {{unordered list
| |
| |1= a sequence of [[random variable]]s <math>\scriptstyle \{X_n\}_{n=1}^\infty</math>, not necessarily sharing a common [[probability space]],
| |
| |2= the sequence of corresponding [[Characteristic function (probability theory)|characteristic function]]s <math>\scriptstyle \{\varphi_n\}_{n=1}^\infty</math>, which by definition are
| |
| : <math>\varphi_n(t) = \operatorname{E}\, e^{itX_n} \quad \forall t\in\mathbb{R},\ \forall n\in\mathbb{N},</math>
| |
| where E is the [[expected value]] operator.
| |
| }}
| |
| | |
| If the sequence of characteristic functions [[pointwise convergence|converges pointwise]] to some function ''φ''
| |
| :<math>\varphi_n(t)\to\varphi(t) \quad \forall t\in\mathbb{R},</math>
| |
| then the following statements become equivalent:
| |
| {{unordered list
| |
| |1= <math>X_n</math> [[Convergence of random variables#Convergence in distribution|converges in distribution]] to some [[random variable]] ''X''
| |
| :<math>X_n\ \xrightarrow{\mathcal D}\ X,</math>
| |
| i.e. the cumulative distribution functions corresponding to random variables converge at every continuity point;
| |
| |2= <math>\scriptstyle \{X_n\}_{n=1}^\infty</math> is [[Tightness of measures|tight]]:
| |
| :<math>\lim_{x\to\infty}\left( \sup_n \operatorname{P}\big[\, |X_n|>x \,\big]\right) = 0;</math>
| |
| |3= ''φ''(''t'') is a characteristic function of some random variable ''X'';
| |
| |4= ''φ''(''t'') is a [[continuous function]] of ''t'';
| |
| |5= ''φ''(''t'') is [[continuous function|continuous]] at ''t'' = 0.
| |
| }}
| |
| | |
| == Proof ==
| |
| Rigorous proofs of this theorem are available.<ref name=W/><ref>Fristedt & Gray (1996, Theorem 18.21)</ref>
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{cite book
| |
| | last = Williams | first = D. | authorlink = David Williams (mathematician)
| |
| | title = Probability with Martingales
| |
| | year = 1991
| |
| | publisher = Cambridge University Press
| |
| | isbn = 0-521-40605-6
| |
| }}
| |
| *Fristedt, B.E.; Gray, L. F. (1996): [http://www.springer.com/birkhauser/applied+probability+and+statistics/book/978-0-8176-3807-8'' A modern approach to probability theory''], Birkhäuser Boston. ISBN 0-8176-3807-5
| |
| | |
| == External links ==
| |
| *[http://ocw.mit.edu/courses/mathematics/18-175-theory-of-probability-fall-2008/lecture-notes/ Lecture notes of "Theory of Probability" from MIT Open Course ] Sessions 9–14 are related to this theorem.
| |
| | |
| {{DEFAULTSORT:Levy continuity theorem}}
| |
| [[Category:Probability theorems]]
| |
| [[Category:Statistical theorems]]
| |
Hello and welcome. My name is Figures Wunder. Since she was 18 she's been working as a receptionist but her promotion by no means comes. South Dakota is exactly where me and my husband reside. The preferred hobby for my children and me is to play baseball and I'm trying to make it a profession.
My web-site healthy food delivery - have a peek at this web-site,