= Nested polygonal approximation of Brownian rough paths
For deterministic nested partitions with mesh tending to zero, lift the polygonal interpolation of <Brownian motion> by its ordinary iterated integrals. These <path signatures> converge almost surely to <enhanced Brownian motion> in every Hölder rough-path metric of exponent below $1/2$, hence in $p$-variation for $p>2$. Conditioning on the partition values gives a nested <martingale> for the first level and the off-diagonal second levels. Moment bounds and interpolation upgrade pointwise convergence to convergence in the <rough path metric>.
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