# 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 WaveletRepresentationOfBrownianMotion 2013-03-22 15:12:51 2013-03-22 15:12:51 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Theorem msc 60J65 construction of Brownian motion