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| {{Redirect|Tail event|"tail events" meaning "rare events"|fat tail}}
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| In [[probability theory]], '''Kolmogorov's zero–one law''', named in honor of [[Andrey Nikolaevich Kolmogorov]], specifies that a certain type of [[Event (probability theory)|event]], called a ''tail event'', will either [[almost surely]] happen or almost surely not happen; that is, the [[probability]] of such an event occurring is zero or one.
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| Tail events are defined in terms of infinite [[sequence]]s of [[random variable]]s. Suppose
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| :<math>X_1,X_2,X_3,\dots\,</math> | |
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| is an infinite sequence of [[statistical independence|independent]] random variables (not necessarily identically distributed). Then, a '''tail event''' is an event whose occurrence or failure is determined by the values of these random variables but which is [[statistical independence|probabilistically independent]] of each finite subset of these random variables. For example, the event that the sequence converges, and the event that its sum converges are both tail events. In an infinite sequence of coin-tosses, a sequence of 100 consecutive heads occurring infinitely many times is a tail event.
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| In many situations, it can be easy to apply Kolmogorov's zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine ''which'' of these two extreme values is the correct one.
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| == Formulation == | |
| A more general statement of Kolmogorov's zero–one law holds for sequences of independent [[sigma algebra|σ-algebra]]s. Let (Ω,''F'',''P'') be a [[probability space]] and let ''F''<sub>''n''</sub> be a sequence of mutually independent σ-algebras contained in ''F''. Let
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| :<math>G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)</math>
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| be the smallest σ-algebra containing ''F''<sub>''n''</sub>, ''F''<sub>''n''+1</sub>, …. Then Kolmogorov's zero–one law asserts that for any event
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| :<math>F\in \bigcap_{n=1}^\infty G_n</math>
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| one has either ''P''(''F'') = 0 or 1.
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| The statement of the law in terms of random variables is obtained from the latter by taking each ''F''<sub>''n''</sub> to be the σ-algebra generated by the random variable ''X''<sub>''n''</sub>. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all ''X''<sub>''n''</sub>, but which is independent of any finite number of ''X''<sub>''n''</sub>. That is, a tail event is precisely an element of the intersection <math>\textstyle{\bigcap_{n=1}^\infty G_n}</math>.
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| ==Examples==
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| An [[invertible]] [[measure-preserving transformation]] on a [[standard probability space]] that obeys the 0-1 law is called a [[Kolmogorov automorphism]]. All [[Bernoulli automorphism]]s are Kolmogorov automorphisms but not ''vice-versa''.
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| ==See also==
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| * [[Borel–Cantelli lemma]]
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| * [[Hewitt–Savage zero–one law]]
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| * [[Lévy's zero–one law]]
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| ==References==
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| *{{Citation | last1=Stroock | first1=Daniel | title=Probability theory: An analytic view | publisher=[[Cambridge University Press]] | edition=revised | isbn=978-0-521-66349-6 | year=1999}}.
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| *{{Citation | first = Zdzislaw | last = Brzezniak | authorlink = Zdzislaw Brzezniak | coauthors = [[Tomasz Zastawniak]] | year = 2000 | title = Basic Stochastic Processes | publisher = [[Springer Science+Business Media|Springer]] | isbn = 3-540-76175-6}}
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| *{{Citation | last1=Rosenthal | first1=Jeffrey S. | title=A first look at rigorous probability theory | publisher=World Scientific Publishing Co. Pte. Ltd. | location=Hackensack, NJ | isbn=978-981-270-371-2 | year=2006 | page=37}}
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| ==External links==
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| * [http://www.kolmogorov.com/ The Legacy of Andrei Nikolaevich Kolmogorov] Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A. N. Kolmogorov. A. N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A. N. Kolmogorov.
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| {{DEFAULTSORT:Kolmogorov's zero-one law}}
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| [[Category:Probability theorems]]
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| [[Category:Covering lemmas]]
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