= Lévy construction of Brownian motion
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{title2=$B_{(u+v)/2}=(B_u+B_v)/2+\sqrt{v-u}\,Z/2$}
= Lévy-Ciesielski construction
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{synonym}
The Lévy construction builds <Brownian motion> on $[0,1]$ by successive independent midpoint displacements on dyadic intervals. Start with $B_0=0$ and $B_1=Z_0$, where $Z_0$ has the <standard normal distribution>. On each dyadic interval $[u,v]$, choose a fresh <independent> <standard normal random variable> $Z$ and use the displayed midpoint rule. Conditional on the endpoints, this is the correct <Brownian bridge> midpoint law.
To verify the increment law, if $D=B_v-B_u\sim N(0,\ell)$ with $\ell=v-u$, the new increments are $D/2\pm\sqrt\ell Z/2$. They are <Jointly Gaussian>, each has <variance> $\ell/2$, and their <covariance> is zero, hence they are <independent>. Induction gives all dyadic <Brownian increments>. The change between consecutive linear interpolants has uniform <norm> $2^{-n/2-1}\max_{k<2^n}|Z_{n,k}|$. The <Gaussian tail bound> and <union bound> bound the chance that this maximum exceeds $n$ by $2^{n+1}e^{-n^2/2}$. The <First Borel-Cantelli lemma> and $\sum_n n2^{-n/2}<\infty$ prove <uniform convergence> <almost surely>. The continuous limit has the required <Brownian increments> at arbitrary times by dyadic approximation.
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