Elliptical distribution: Difference between revisions

Revision as of 20:46, 25 December 2013

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 inclusion of the $σ$ -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the $σ$ -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 $σ$ -algebra that is generated by the random variable.

Statement of the lemma

Let $Ω$ be a sample space. For a function $f : Ω \to R^{n}$ , the $σ$ -algebra generated by $f$ is defined as the family of sets $f^{- 1} (S)$ , where $S$ are all Borel sets.

Lemma Let $X, Y : Ω \to R^{n}$ be random elements and $σ (X)$ be the $σ$ algebra generated by $X$ . Then $Y$ is $σ (X)$ -measurable if and only if $Y = g (X)$ for some Borel measurable function $g : R^{n} \to R^{n}$ .

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, $Y$ being $σ (X)$ -measurable is the same as $Y^{- 1} (S) \in σ (X)$ for any Borel set $S$ , which is the same as $σ (Y) \subset σ (X)$ . So, the lemma can be rewritten in the following, equivalent form.

Lemma Let $X, Y : Ω \to R^{n}$ be random elements and $σ (X)$ and $σ (Y)$ the $σ$ algebras generated by $X$ and $Y$ , respectively. Then $Y = g (X)$ for some Borel measurable function $g : R^{n} \to R^{n}$ if and only if $σ (Y) \subset σ (X)$ .

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

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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.
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+==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]]

Elliptical distribution: Difference between revisions

Revision as of 20:46, 25 December 2013

Statement of the lemma

References

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