Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 201 / 4 / b / Solution

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.