TITLE: Weierstrass' function and Brownian motion
QUESTION [5 upvotes]: Is there a known connection between Weierstrass' function 
$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$  
and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass function has same Holder continuity peoperties that Brownain sample paths do. Some crude Matlab experiments seem to suggest this function has linear quadratic variation for low values of $b$, too.
The expression reminds me a little of the Karhunen-loeve expansion of Brownian motion, but I don't see how the two might relate.
Many thanks.

REPLY [3 votes]: Some quick Googling brought me to this paper. The idea is to take the coefficients in the summation to be suitable independent random variables according to a suggestion of Mandelbrot. I can't actually access the paper right now so I can't say if the authors were able to include the case of standard Brownian motion in their results. 
``Convergence of the Weierstrass-Mandelbrot process to Fractional Brownian Motion'' Murad Taqqu and Vladas Pipiras.  Fractals. 8 (2000) 369-384.

REPLY [3 votes]: You might find this account useful. In particular, see the end of Section 1, page 7, and the end of Section 4.