 sequences of random variables $$\begin{aligned} &&n^{\beta-0}\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=0}^k\theta_i\hat {\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta\quad \mbox{and}\\ &&n^{\beta-0}\sum_{x\in\mathbb{Z}} \Biggl( \Biggl|\sum_{i=0}^k\theta_i\hat {\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta \operatorname{sgn} \Biggl(\sum_{i=0}^k\theta_i\hat{\Gamma}_n(\tau_i,\{x/n\}) \Biggr) \Biggr)\end{aligned}$$ converge jointly in distribution toward the respective random variables $$\int_{-\infty}^\infty\Biggl|\sum_{i=0}^k\theta_iL_\ast(\tau_i,x) \Biggr|^\beta \,\mathrm{d}x \quad \mbox{and}\quad \int_{-\infty}^\infty\Biggl( \Biggl|\sum_{i=0}^k\theta_iL_\ast(\tau_i,x) \Biggr|^\beta \operatorname{sgn} \Biggl(\sum_{i=0}^k\theta_iL_\ast(\tau_i,x) \Biggr) \Biggr)\,\mathrm{d}x .$$ 