Hausdorff dimension of Weierstrass-Mandelbrot process
Let \(W_r(t)= \sqrt{(\ln r)}\sum _{n=-\infty} ^{n=\infty} (1-\exp(ir^nt)) r^{-nH} \eta_n\), where \(\eta_n= \exp(i\theta_n)\), \( \theta_n\) are i.i.d. uniform variables on \([0,2\pi]\), \(H\in (0,1)\) and \(r\in (1,\infty)\). Such process is usually called Weierstrass-Mandelbrot process and as \(r\to 1\), \(W_r(t)\) converges in distribution to a complex fractional Brownian motion with the Hurst parameter equal to \(H\). The author proves that Weierstrass-Mandelbrot process and fractional Brownian motion have the same Hausdorff dimension of path and ranges, and, thus, verify the Berry-Lewis hypothesis [\textit{M. V. Berry} and \textit{Z. V. Lewis}, Proc. R. Soc. Lond., Ser. A 370, 459-484 (1980; Zbl 0435.28008)].
scientific article; zbMATH DE number 1796691