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The $2^n$ increments are independent normal random variables, each distributed as $2^{-n/2}{B}_1$. Their squared means are $2^{-n}$ and, using the permitted variance, their squared variances are $2\cdot 2^{-2n}$. Independence therefore givesConsequently the dyadic quadratic variation converges to one:This is dyadic quadratic variation of Brownian motion on the time interval of length one. The calculation requires independence only within each partition; increments from different resolutions need not be independent.

A function with finite Hölder seminorm of exponent $\alpha >1/2$ has squared-increment sum on a uniform $N$-interval partition bounded by $C^2{N}^{1-2\alpha }$, hence tending to zero. This conflicts with a nonzero quadratic-variation limit. In particular dyadic quadratic variation of Brownian motion prevents a Brownian coordinate law supported on such a Hölder space. This argument alone gives no conclusion at the critical exponent $1/2$.