en>Limit-theorem: /* Definition */

2014-08-03T18:38:38Z

Definition

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	In [[mathematics]], in particular [[probability theory]], the '''Doob–Dynkin lemma''', named after [[Joseph L. Doob]] and [[Eugene Dynkin]], characterizes the situation when one [[random variable]] is a function of another by the [[subset\|inclusion]] of the [[sigma algebra\|<~~math~~>~~\sigma~~<~~/math~~>-~~algebras]] generated by~~ the ~~random variables~~. ~~The usual statement~~ of ~~the lemma~~ is ~~formulated in terms of one random variable being [[measurable function\|measurable]]~~ with ~~respect~~ to the <~~math~~>~~\sigma~~</~~math>-algebra generated by the other~~.		<br><br>This should-have kit capabilities a belt sander that won the Handy Magazine Innovation Award in 2006 for its host of winning options. Ridgid it got it right with this one particular! Thus, you can install your 18 Volt 1/2 Cordless Impact Wrench belt on your belt sander devoid of any concern more than which path your Fintech belt is rotating. 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	~~The lemma plays an important role in the [[conditional expectation~~]~~] in probability theory, where~~ it ~~allows~~ to ~~replace~~ the ~~conditioning on~~ a ~~[[random variable]]~~ by ~~conditioning on the [[sigma-algebra\|<math>\sigma</math>-algebra]] that is [[Sigma_algebra#~~.CF.~~83-algebra_generated_by_a_function\|generated]] by the random~~ variable.

	~~==Statement of~~ the ~~lemma==~~
	~~Let~~ <~~math~~>~~\Omega~~<~~/math~~> be a ~~[[sample space]]~~. ~~For a function <math>f:\Omega \rightarrow R^n</math>~~, the ~~<math>\sigma</math>-algebra generated by <math>f</math>~~ is ~~defined~~ as ~~the family of sets~~ <~~math~~>~~f^{-1}(S)~~<~~/math~~>, ~~where <math>S</math>~~ are ~~all [[Borel set]]s~~.

	'''~~Lemma''' Let <math>X,Y: \Omega \rightarrow R^n</math> be [[random element]]~~s ~~and <math>\sigma(X)</math>~~ ~~be the <math>\sigma</math> algebra generated by <math>X</math>. Then <math>Y</math> is <math>\sigma(X)</math>-[[measurable]] if and only if <math>Y=g(X)</math>~~ for ~~some [[Borel measurable]] function <math>g:R^n\rightarrow R^n</math>~~.

	The ~~"if" part~~ of the ~~lemma is simply~~ the ~~statement that~~ the ~~composition of two measurable functions is measurable. The "only if" part~~ is the ~~nontrivial one~~.

	~~By definition,~~ <~~math~~>Y</~~math> being <math>\sigma(X)<~~/~~math>~~-~~[[measurable~~]~~] is~~ the ~~same~~ as ~~<math>Y^{-1}(S)\~~in ~~\sigma(X)</math>~~ for any ~~Borel set~~ <~~math~~>S<~~/math~~>, ~~which is the same as <math>\sigma(Y) \subset \sigma(X)</math>~~. So, ~~the lemma can be rewritten~~ in ~~the following, equivalent form~~.

	~~'''Lemma''' Let <math>X~~,~~Y: \Omega \rightarrow R^n</math>~~ be ~~random elements and~~ ~~<math>\sigma(X)</math>~~ and <~~math~~>~~\sigma(Y)~~<~~/math~~> the ~~<math>\sigma</math> algebras generated~~ by ~~<math>X</math>~~ and <~~math~~>Y<~~/math~~>, ~~respectively. Then <math>Y=g(X)</math> for some Borel measurable function <math>g:R^n\rightarrow R^n</math> if~~ and ~~only if <math>\sigma(Y) \subset \sigma(X)</math>~~.

	~~==References==~~
	* A. ~~Bobrowski: ''Functional analysis for probability and stochastic processes: an introduction'', Cambridge University Press~~ (~~2005~~)~~, ISBN 0-521-83166-0~~
	* M. ~~M. Rao, R. J~~. ~~Swift : ''Probability Theory with Applications'', Mathematics and Its Applications, Band 582, Springer-Verlag (2006), ISBN 0-387-27730-7~~

	~~{{DEFAULTSORT:Doob-Dynkin Lemma}}~~
	~~[[Category:Probability theorems]]~~
	~~[[Category:Measure theory]]~~

en>Ultimate cosmic evil: /* Applications */

2013-12-25T19:46:33Z

Applications

← Older revision		Revision as of 21:46, 25 December 2013
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			In [[mathematics]], in particular [[probability theory]], the '''Doob–Dynkin lemma''', named after [[Joseph L. Doob]] and [[Eugene Dynkin]], characterizes the situation when one [[random variable]] is a function of another by the [[subset\|inclusion]] of the [[sigma algebra\|<math>\sigma</math>-algebras]] generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being [[measurable function\|measurable]] with respect to the <math>\sigma</math>-algebra generated by the other.

			The lemma plays an important role in the [[conditional expectation]] in probability theory, where it allows to replace the conditioning on a [[random variable]] by conditioning on the [[sigma-algebra\|<math>\sigma</math>-algebra]] that is [[Sigma_algebra#.CF.83-algebra_generated_by_a_function\|generated]] by the random variable.

	~~We should find some to test drive~~ a ~~rental truck to~~ [~~http~~://~~www~~.~~bbc~~.~~co.uk~~/~~search~~/?q=~~trickle trickle~~] ~~charge~~ the ~~batteries For more regarding~~ [~~http~~://~~miamiusedcars~~.~~org~~/ ~~Used Cars St Louis Craigslist] stop by our web page~~. .		==Statement of the lemma==
			Let <math>\Omega</math> be a [[sample space]]. For a function <math>f:\Omega \rightarrow R^n</math>, the <math>\sigma</math>-algebra generated by <math>f</math> is defined as the family of sets <math>f^{-1}(S)</math>, where <math>S</math> are all [[Borel set]]s.

			'''Lemma''' Let <math>X,Y: \Omega \rightarrow R^n</math> be [[random element]]s and <math>\sigma(X)</math> be the <math>\sigma</math> algebra generated by <math>X</math>. Then <math>Y</math> is <math>\sigma(X)</math>-[[measurable]] if and only if <math>Y=g(X)</math> for some [[Borel measurable]] function <math>g:R^n\rightarrow R^n</math>.

			The "if" part of the lemma is simply the statement that the composition of two measurable functions is measurable. The "only if" part is the nontrivial one.

			By definition, <math>Y</math> being <math>\sigma(X)</math>-[[measurable]] is the same as <math>Y^{-1}(S)\in \sigma(X)</math> for any Borel set <math>S</math>, which is the same as <math>\sigma(Y) \subset \sigma(X)</math>. So, the lemma can be rewritten in the following, equivalent form.

			'''Lemma''' Let <math>X,Y: \Omega \rightarrow R^n</math> be random elements and <math>\sigma(X)</math> and <math>\sigma(Y)</math> the <math>\sigma</math> algebras generated by <math>X</math> and <math>Y</math>, respectively. Then <math>Y=g(X)</math> for some Borel measurable function <math>g:R^n\rightarrow R^n</math> if and only if <math>\sigma(Y) \subset \sigma(X)</math>.

			==References==
			* A. Bobrowski: ''Functional analysis for probability and stochastic processes: an introduction'', Cambridge University Press (2005), ISBN 0-521-83166-0
			* M. M. Rao, R. J. Swift : ''Probability Theory with Applications'', Mathematics and Its Applications, Band 582, Springer-Verlag (2006), ISBN 0-387-27730-7

			{{DEFAULTSORT:Doob-Dynkin Lemma}}
			[[Category:Probability theorems]]
			[[Category:Measure theory]]

en>Mathstat: /* References */ Symmetric Multivariate and Related Distributions

2012-06-06T16:57:34Z

References: Symmetric Multivariate and Related Distributions

New page

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Elliptical distribution - Revision history

en>Limit-theorem: /* Definition */

en>Ultimate cosmic evil: /* Applications */

en>Mathstat: /* References */ Symmetric Multivariate and Related Distributions