Covariance matrix - Revision history

en>Loraof: /* Correlation matrix */ sizes of the entries

2014-12-27T01:13:45Z

Correlation matrix: sizes of the entries

← Older revision		Revision as of 03:13, 27 December 2014
Line 1:		Line 1:
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130.233.176.27: The source has been incorrectly refered - (covariance matrix part was found by Higham under different details already in 1988)

2014-02-20T09:35:47Z

The source has been incorrectly refered - (covariance matrix part was found by Higham under different details already in 1988)

← Older revision		Revision as of 11:35, 20 February 2014
Line 1:		Line 1:
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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 (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~~
	~~:<math>G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)</math>~~
	~~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~~
	~~:<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 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>.~~

	~~==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]~~]

82.250.15.56: /* Which matrices are covariance matrices? */ No need for qu/ans game. Just make the statement

2013-12-11T17:45:24Z

Which matrices are covariance matrices?: No need for qu/ans game. Just make the statement

← Older revision		Revision as of 19:45, 11 December 2013
Line 1:		Line 1:
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	[~~http://Www~~.~~Ehow~~.~~com~~/~~search~~.~~html?s=Truthfully Truthfully~~] ~~Supercell~~, ~~by allowing currently~~ the ~~illusion on~~ the ~~multi player game~~, ~~taps into your instinctual male drive as~~ a ~~way to from the status hierarchy~~, ~~and even though~~ it''s ~~unattainable~~ to the ~~surface~~ of ~~your hierarchy if there isn~~'~~t been logging in every single because the game turned out plus you invested actually money in extra builders~~, ~~the drive for obtaining~~ a ~~small bit further forces enough visitors to use up~~ a ~~real income~~ in ~~relation to virtual~~ '~~gems~~'~~" that pastime could~~ be the ~~top~~-~~grossing app within the Practical application Store.~~<br><br>~~Temperance is essential in most things~~, ~~and enjoying clash of clans cheats is no some other. Playing for hours on finish is not just good-for-you~~, ~~bodily or at your inner levels. There are some games out typically there which know this and may include measures to remind in order to take rests~~. ~~Go ahead and take initiative yourself, although! Place an alarm so you don~~'~~t play~~ for ~~a lot more an hour right.~~<br><br>~~Look at note of how plenty money your teen is normally shelling out for gaming~~. ~~These kinds~~ of ~~products aren't cheap coupled with then there is very much~~ the ~~option~~ of ~~purchasing much more add-ons near the game itself. Establish month-to-month and to choose~~ from ~~restrictions on~~ the ~~total number of money that is likely~~ to be ~~spent on video. Also, have conversations that has~~ the ~~youngsters about factoring.~~<br><br>A ~~blog~~ is ~~offering Clash associated~~ with ~~Clans hack tool trainer~~ to ~~users who want. The website offering this tool is protected and it guarantees finist quality software. There furthermore other sites which secure~~ the ~~tool. But ranks~~, which ~~indicates either incomplete or~~ of ~~bad quality. when users download these just a few hack tools, instead~~ of ~~doing well they end in in trouble~~. So, ~~players are advised pick the tool from~~ a ~~website that offers complete programs.Users who are finding it tough to fold the hurdles can find a site that allows tourists to download the cheats. Most~~ of the ~~world wide web allow free download along with many websites charge fees. Users can locate a website from where they can obtain good quality software.~~<br><br>~~Should you perform online multi~~-~~player game titles, don't neglect~~ the ~~strength of toned of voice chat! A mic or wireless headset~~ is a ~~very simple expenditure~~, and ~~having the capability to speak to finally your fellow athletes has a lot~~ of ~~success~~. ~~You are allowed to create more influential connections with the spot the community and stay your far more successful institution person when you can be able connect out obnoxious~~.~~<br><br>Now you have read this composition~~, ~~you need to have an easier time locating then loving video games in your own life. Notwithstanding your favored platform~~, ~~from your cellphone on the own computer~~, ~~playing furthermore enjoying video gaming allows for you to take the advantage~~ of ~~the worries of your favorite busy week get advise~~.		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\|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
			:<math>G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)</math>
			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
			:<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 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>.

			==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]]

en>Prax54: /* Which matrices are covariance matrices? */

2012-08-12T18:30:11Z

Which matrices are covariance matrices?

New page

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