wavelet representation of Brownian motion

First we define the function

$H(t)=\begin{cases}1&\text{for }0\leq t<\tfrac{1}{2}\\ -1&\text{for }\tfrac{1}{2}\leq t\leq 1\\ 0&\text{otherwise.}\end{cases}$

(1)

and the sequence of functions

$H_{n}(t)=2^{j/2}H(2^{j}t-k)$

(2)

for $n=2^{j}+k$ where $j>0$ and $0\leq k\leq 2^{j}$ . We also set $H_{0}(t)=1$ .

Wavelet Representation of Brownian Motion.

If $\{Z_{n}:0\leq n<\infty\}$ is a sequence of independent Gaussian random variables with mean $0$ and variance $1$ , then the series defined by

$X_{t}=\sum^{\infty}_{n=0}\left(Z_{n}\int_{0}^{t}H_{n}(s)\;ds\right)$

(3)

converges uniformly on $[0,1]$ with probability one. Moreover, the process $\{X_{t}\}$ defined by the limit is a Brownian motion for $0\leq t\leq 1$ .

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