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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.
 
Tail events are defined in terms of infinite [[sequence]]s of [[random variable]]s. Suppose
 
:<math>X_1,X_2,X_3,\dots\,</math>
 
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.
 
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.
 
== Formulation ==
A more general statement of Kolmogorov's zero–one law holds for sequences of independent [[sigma algebra|&sigma;-algebra]]s. Let (&Omega;,''F'',''P'') be a [[probability space]] and let ''F''<sub>''n''</sub> be a sequence of mutually independent &sigma;-algebras contained in ''F''. Let
:<math>G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)</math>
be the smallest &sigma;-algebra containing ''F''<sub>''n''</sub>, ''F''<sub>''n''+1</sub>, &hellip;. Then Kolmogorov's zero–one law asserts that for any event
:<math>F\in \bigcap_{n=1}^\infty G_n</math>
one has either ''P''(''F'') = 0 or 1.
 
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 &sigma;-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 &sigma;-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>.
 
==Examples==
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''.
 
==See also==
* [[Borel–Cantelli lemma]]
* [[Hewitt–Savage zero–one law]]
* [[Lévy's zero–one law]]
 
==References==
*{{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}}.
*{{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}}
*{{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}}
 
==External links==
* [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.
 
{{DEFAULTSORT:Kolmogorov's zero-one law}}
[[Category:Probability theorems]]
[[Category:Covering lemmas]]

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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, 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.

Tail events are defined in terms of infinite sequences of random variables. Suppose

X1,X2,X3,

is an infinite sequence of 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 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.

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.

Formulation

A more general statement of Kolmogorov's zero–one law holds for sequences of independent σ-algebras. Let (Ω,F,P) be a probability space and let Fn be a sequence of mutually independent σ-algebras contained in F. Let

Gn=σ(k=nFk)

be the smallest σ-algebra containing Fn, Fn+1, …. Then Kolmogorov's zero–one law asserts that for any event

Fn=1Gn

one has either P(F) = 0 or 1.

The statement of the law in terms of random variables is obtained from the latter by taking each Fn to be the σ-algebra generated by the random variable Xn. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all Xn, but which is independent of any finite number of Xn. That is, a tail event is precisely an element of the intersection n=1Gn.

Examples

An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism. All Bernoulli automorphisms are Kolmogorov automorphisms but not vice-versa.

See also

References

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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External links

  • 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.