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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]]

Latest revision as of 02:13, 27 December 2014

How to get rid of weight cycling? An inside depth guide into how to lose fat cycling? Use my truly effective how to lose fat cycling programme completely free. I'll show you how I lost more than 2.5 stone cycling in really 9 weeks, I'm going to tell you how I did it, so to benefit and get down to your perfect fat.

Food is the fuel of the body and you have to understand which though fasting is included inside the extended list of ways to get rid of fat it's at the finish of the day a very unhealthy practice. Instead of getting oily and spiced up food stuffs you'll need to commence the usage of cereals plus food items which have a higher fiber and grain content. One of the better ways to lose weight fast is the consumption of 2 servings of fruit and 5 servings of veggies a day.

One last question: where does fat disappear first plus last? When you lose weight by doing exercises, we lose fat initially in the trunk (shoulders, back and abdomen) and upper limbs (arms especially). Then finally the buttocks and thighs.

More recently, certain businesses are making it even easier for a breastfeeding mother to commence losing fat. Slim Fast plus Weight Watchers have a specialized diet merely for nursing moms. The South Beach Diet can moreover be employed should you simply skip over Phase One of the program. Many diets have now altered their plans to fit a nursing mother's needs.

And whenever I am seized with all the temptation of the wicked doughnut, I will resist-at least for a time how to lose weight fast. And then another time, till I've put enough time between the impulse plus the action, plus the wanting has subsided.

Exercising initially thing in the morning eliminates these feelings. You have finished before the day begins. We will be happier because you'll understand that the rest of the day is free considering you will have already completed what we needed to. There can be no guilt plus no dreading exercise considering it will already be completed.

You are able to follow different kinds of diets like Atkins diet, low-carb diet, high-protein diet, etc., and include the above-mentioned foods in your diet plan to gain maximum benefits. All the best with a endeavor!

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