HOME

TheInfoList



Click Here for Items Related To - Lévy's Modulus Of Continuity Theorem
OR:
Lévy's Modulus Of Continuity Theorem on:   [Wikipedia]   [Google]   [Amazon]

Lévy's modulus of continuity theorem is a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
that gives a result about an
almost sure In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
behaviour of an estimate of the
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f ...
for
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
, that is used to model what's known as Brownian motion. Lévy's modulus of continuity theorem is named after the French mathematician Paul Lévy.


Statement of the result

Let B :
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
\times \Omega \to \mathbb be a standard Wiener process. Then,
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
, :\lim_ \sup_ \frac = 1. In other words, the sample paths of Brownian motion have modulus of continuity :\omega_ (\delta) = \sqrt with probability one, and for sufficiently small \delta > 0.Lévy, P. Author Profile Théorie de l’addition des variables aléatoires. 2. éd. (French) page 172 Zbl 0056.35903 (Monographies des probabilités.) Paris: Gauthier-Villars, XX, 387 p. (1954)


See also

* Some properties of sample paths of the Wiener process


References

* Paul Pierre Lévy, ''Théorie de l'addition des variables aléatoires.'' Gauthier-Villars, Paris (1937). {{DEFAULTSORT:Levy's modulus of continuity theorem Probability theorems
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...