still true that $f$ is constant. This can be proven by martingale theory, as can the statement that harmonic functions bounded below are constant ( Liouville's theorem ). Let $X_1,X_2,\ldots$ be a sequence of independent random random variables uniformly distributed on the unit circle, set $S_n=\sum_{m=1}^nX_m$ and let $\mathcal{F}_n$ be the sigma-algebra