wavelet representation of Brownian motion

First we define the function

(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 $0\le k\le 2^j$. We also set $H_0(t)=1$.

Wavelet Representation of Brownian Motion.

If is a sequence of independent Gaussian random variables with mean$\mathrm{0}$ and variance $\mathrm{1}$, then the series defined by

$$X_t=\sum _{n=0}^{\mathrm{\infty }}\left(Z_n{\int }_0^t{H}_n(s)𝑑s\right)$$(3)

converges uniformly on $\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{]}$ with probability one. Moreover, the process $\mathrm{\{}{X}_t\mathrm{\}}$ defined bythe limit is a Brownian motion for $\mathrm{0}\mathrm{\le }t\mathrm{\le }\mathrm{1}$.