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| | Nice to meet you, my title is Ling and I totally dig that name. Managing people is what I do in my working day occupation. I currently live in Alabama. What she loves performing is taking part in croquet and she is attempting to make it a profession.<br><br>Here is my web blog :: extended auto warranty; [http://www.Drlouiseboothby.com/UserProfile/tabid/660/userId/8985/Default.aspx straight from the source], |
| In [[mathematics]], '''finite-dimensional distributions''' are a tool in the study of [[Measure theory|measures]] and [[stochastic processes]]. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional [[vector space]] (or finite collection of times).
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| ==Finite-dimensional distributions of a measure==
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| Let <math>(X, \mathcal{F}, \mu)</math> be a [[measure space]]. The '''finite-dimensional distributions''' of <math>\mu</math> are the [[pushforward measure]]s <math>f_{*} (\mu)</math>, where <math>f : X \to \mathbb{R}^{k}</math>, <math>k \in \mathbb{N}</math>, is any measurable function.
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| ==Finite-dimensional distributions of a stochastic process==
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| Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[probability space]] and let <math>X : I \times \Omega \to \mathbb{X}</math> be a [[stochastic process]]. The '''finite-dimensional distributions''' of <math>X</math> are the push forward measures <math>\mathbb{P}_{i_{1} \dots i_{k}}^{X}</math> on the [[product space]] <math>\mathbb{X}^{k}</math> for <math>k \in \mathbb{N}</math> defined by
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| :<math>\mathbb{P}_{i_{1} \dots i_{k}}^{X} (S) := \mathbb{P} \left\{ \omega \in \Omega \left| \left( X_{i_{1}} (\omega), \dots, X_{i_{k}} (\omega) \right) \in S \right. \right\}.</math>
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| Very often, this condition is stated in terms of [[measurable]] [[rectangle]]s:
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| :<math>\mathbb{P}_{i_{1} \dots i_{k}}^{X} (A_{1} \times \cdots \times A_{k}) := \mathbb{P} \left\{ \omega \in \Omega \left| X_{i_{j}} (\omega) \in A_{j} \mathrm{\,for\,} 1 \leq j \leq k \right. \right\}.</math>
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| The definition of the finite-dimensional distributions of a process <math>X</math> is related to the definition for a measure <math>\mu</math> in the following way: recall that the [[Law (stochastic processes)|law]] <math>\mathcal{L}_{X}</math> of <math>X</math> is a measure on the collection <math>\mathbb{X}^{I}</math> of all functions from <math>I</math> into <math>\mathbb{X}</math>. In general, this is an infinite-dimensional space. The finite dimensional distributions of <math>X</math> are the push forward measures <math>f_{*} \left( \mathcal{L}_{X} \right)</math> on the finite-dimensional product space <math>\mathbb{X}^{k}</math>, where
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| :<math>f : \mathbb{X}^{I} \to \mathbb{X}^{k} : \sigma \mapsto \left( \sigma (t_{1}), \dots, \sigma (t_{k}) \right)</math>
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| is the natural "evaluate at times <math>t_{1}, \dots, t_{k}</math>" function.
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| ==Relation to tightness==
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| It can be shown that if a sequence of [[probability measure]]s <math>(\mu_{n})_{n = 1}^{\infty}</math> is [[Tightness of measures|tight]] and all the finite-dimensional distributions of the <math>\mu_{n}</math> [[Weak convergence of measures|converge weakly]] to the corresponding finite-dimensional distributions of some probability measure <math>\mu</math>, then <math>\mu_{n}</math> converges weakly to <math>\mu</math>.
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| ==See also==
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| * [[Law (stochastic processes)]]
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| {{DEFAULTSORT:Finite-Dimensional Distribution}}
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| [[Category:Measure theory]]
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| [[Category:Stochastic processes]]
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Nice to meet you, my title is Ling and I totally dig that name. Managing people is what I do in my working day occupation. I currently live in Alabama. What she loves performing is taking part in croquet and she is attempting to make it a profession.
Here is my web blog :: extended auto warranty; straight from the source,