wavelet representation of Brownian motion

First we define the function

H(t)={1for 0t<12-1for 12t10otherwise. (1)

and the sequence of functions

Hn(t)=2j/2H(2jt-k) (2)

for n=2j+k where j>0 and 0k2j. We also set H0(t)=1.

Wavelet Representation of Brownian Motion.

If {Zn:0n<} is a sequence of independentPlanetmathPlanetmath Gaussian random variablesMathworldPlanetmath with mean 0 and variance 1, then the series defined by

Xt=n=0(Zn0tHn(s)𝑑s) (3)

converges uniformly on [0,1] with probability one. Moreover, the process {Xt} defined by the limit is a Brownian motionMathworldPlanetmath for 0t1.

Title wavelet representation of Brownian motion
Canonical name WaveletRepresentationOfBrownianMotion
Date of creation 2013-03-22 15:12:51
Last modified on 2013-03-22 15:12:51
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Theorem
Classification msc 60J65
Synonym construction of Brownian motion