Given $a,b>0$, let $f_n:\mathbb{R}\to\mathbb{R}$ for $n>0$ be the random variable $t\mapsto(1/\sqrt{n})\sum_{k<n}\sin(t/(a+kb/n)+\phi_k)$ where $\phi$ is uniformly distributed in $[0,2\pi)^n$.
Is there some sort of convergence in this sequence, maybe weak convergence in probability to a limit variable $f:\mathbb{R}\to\mathbb{R}$?
What if $t\mapsto(1/\sqrt{n})\sum_{k<n}\sin((a+kb/n)t+\phi_k)$ (equidistant frequencies vs. equidistant wavelengths)?