Epsilon Eridani in fiction: Difference between revisions

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In [[mathematics]], '''Lévy's modulus of continuity theorem''' gives a result about the [[almost surely|almost sure]] behaviour of an estimate of the [[modulus of continuity]] for the [[Wiener process]], which models [[Brownian motion]]. It is due to the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]].
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==Statement of the result==
 
Let <math>B : [0, 1] \times \Omega \to \mathbb{R}</math> be a standard Wiener process. Then, [[almost surely]],
 
:<math>\lim_{h \to 0} \sup_{0 \leq t \leq 1 - h} \frac{| B_{t+ h} - B_{t} |}{\sqrt{2 h \log (1 / h)}} = 1.</math>
 
In other words, the [[sample path]]s of Brownian motion have modulus of continuity
 
:<math>\omega_{B} (\delta) = \sqrt{2 \delta \log (1 / \delta)}</math>
 
with probability one, and for sufficiently small <math>\delta > 0</math>.
 
==See also==
* [[Wiener process#Some properties of sample paths|Some properties of sample paths of the Wiener process]]
 
==References==
 
* P.P. Lévy. ''Théorie de l'addition des variables aléatoires.'' Gauthier-Villars, Paris (1937).
 
{{DEFAULTSORT:Levy's modulus of continuity theorem}}
[[Category:Probability theorems]]

Latest revision as of 21:58, 25 December 2014

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