= Lévy-Ciesielski expansion
{c}
{title2=$B_t=Z_0t+\sum_{n,k}Z_{n,k}\int_0^tH_{n,k}(s)\,ds$}
Let $H_{n,k}(s)=2^{n/2}(\mathbf1_{(0,1/2)}-\mathbf1_{(1/2,1)})(2^ns-k)$ for $n\geq0$ and $0\leq k<2^n$. 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>.
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