Lévy-Ciesielski expansion ($B_t=Z_{0}t+{\sum }_{n,k}{Z}_{n,k}{\int }_0^t{H}_{n,k}(s)ds$)

Let $H_{n,k}(s)=2^{n/2}(1_{(0,1/2)}-1_{(1/2,1)})(2^{n}s-k)$ for $n\ge 0$ and $0\le k<2^n$. Integrating these Haar wavelets gives tents of height $2^{-n/2-1}$ supported on the corresponding dyadic intervals. For independent standard normal random variables $Z_0,Z_{n,k}$, the displayed series, summed level by level, is exactly the succession of midpoint displacements in the Lévy construction of Brownian motion. Its almost sure convergence is uniform, and its limit is standard Brownian motion. This gives a continuous function from Gaussian coefficients even though the unintegrated unit-variance Haar wavelet series diverges in L2 space.