Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 31 / 6 / c / Solution

There is no density; scaling by two gives a law singular to Wiener measure. On $[0,1]$ form the pathwise dyadic sumsUnder Wiener measure, independent normal increments give $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Chebyshev's inequality and the Borel-Cantelli lemma show $Q_n\to 1$ almost surely, because the variances are summable. For $2B$, every sum is four times its Brownian counterpart and hence tends to four almost surely. The event that the limit equals one separates the two laws. Unlike the linear-drift case, this obstruction already occurs on a finite interval: it is the Pathwise quadratic variation distinguishes Brownian speeds argument.