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| | Greetings. The writer's name is Phebe and she feels comfy when individuals use the complete title. Playing baseball is the pastime he will never quit performing. South Dakota is her beginning place but she needs to move simply because of her family. I am a meter reader but I plan on changing it.<br><br>Look into my blog ... [http://www.sddch.org/?document_srl=345265 http://www.sddch.org] |
| In [[mathematics]], the '''law''' of a [[stochastic process]] is the [[Measure (mathematics)|measure]] that the process induces on the collection of [[Function (mathematics)|functions]] from the [[index set]] into the state space. The law encodes a lot of information about the process; in the case of a [[random walk]], for example, the law is the [[probability measure|probability distribution]] of the possible trajectories of the walk.
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| ==Definition==
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| Let (Ω, ''F'', '''P''') be a [[probability space]], ''T'' some index set, and (''S'', Σ) a [[measurable space]]. Let ''X'' : ''T'' × Ω → ''S'' be a stochastic process (so the map
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| :<math>X_{t} : \Omega \to S : \omega \mapsto X (t, \omega)</math>
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| is a (''F'', Σ)-[[measurable function]] for each ''t'' ∈ ''T''). Let ''S''<sup>''T''</sup> denote the collection of all functions from ''T'' into ''S''. The process ''X'' (by way of [[currying]]) induces a function Φ<sub>''X''</sub> : Ω → ''S''<sup>''T''</sup>, where
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| :<math>\left( \Phi_{X} (\omega) \right) (t) := X_{t} (\omega).</math>
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| The '''law''' of the process ''X'' is then defined to be the [[pushforward measure]]
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| :<math>\mathcal{L}_{X} := \left( \Phi_{X} \right)_{*} ( \mathbf{P} ) = \mathbf P \circ \Phi_X^{-1}</math> | |
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| on ''S''<sup>''T''</sup>.
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| ==Example==
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| * The law of standard [[Brownian motion]] is [[classical Wiener measure]]. (Indeed, many authors define Brownian motion to be a [[sample continuous process]] starting at the origin whose law is Wiener measure, and then proceed to derive the independence of increments and other properties from this definition; other authors prefer to work in the opposite direction.)
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| ==See also==
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| * [[Finite-dimensional distribution]]
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| * [[stochastic process]]
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| {{DEFAULTSORT:Law (Stochastic Processes)}}
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| [[Category:Stochastic processes]]
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| {{probability-stub}}
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