Theorem by Paley-Wiener-Zygmund

The Paley-Wiener-Zygmund theorem is a mathematical theorem from probability theory that deals with the properties of the paths of the Wiener process . This stochastic process forms, among other things, the basis for the modeling of share prices and Brownian molecular movement .

The sentence was published in 1933 by Raymond Paley , Norbert Wiener and Antoni Zygmund , but just in case . ${\displaystyle \gamma =1}$

statement

For each of them , the paths of the Wiener process are almost certainly not locally Hölder-continuous of the order at any point and consequently also almost certainly nowhere differentiable. ${\displaystyle \gamma \in ({\tfrac {1}{2}},\infty )}$ ${\displaystyle \gamma }$

swell

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 477-478 , doi : 10.1007 / 978-3-642-36018-3 .
• Payley Raymond EAC, Wiener Norbert, Zygmund Antoni: Note on random functions . In: Mathematical Journal . No. 37 , 1933, pp. 647–668 ( uni-goettingen.de [accessed October 5, 2016]).