On the Hausdorff dimension of graph of random vector-valued Weierstrass function

Let $\mathrm{\Theta }=\{{\theta }_n\},\mathrm{\Lambda }=\{{\lambda }_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by $$f_{\mathrm{\Theta },\mathrm{\Lambda }}(x)=(\sum _{n=0}^{\mathrm{\infty }}{a}^n\cos (2\pi (b^{n}x+{\theta }_n)),\sum _{n=0}^{\mathrm{\infty }}{a}^n\sin (2\pi (b^{n}x+{\lambda }_n)\left)\right),x\in [0,1],$$ where $0<a<1<b,ab>1$. The Hausdorff dimension of the graph of this function is proved to be $${\dim }_{H}G(f_{\mathrm{\Theta },\mathrm{\Lambda }})=\min \{-\frac{\log b}{\log a},3+2\frac{\log a}{\log b}\}\text{a.s.}$$