Random process - autocorrelation

I am currently working on random processes. Let's consider the random process defined as $$u^s(x,t) = 2\sum_{n=1}^{N} \hat{u}^n \cos(\kappa^n\cdot x + \psi_n + \omega_n t )\sigma^n$$ where $$\hat{u}^n$$, $$\psi_n$$, $$\omega_n$$ and $$\sigma^n$$ are respectivly the amplitude, phase, pulsation and the direction of the fourier mode $$\kappa^n$$.

The time correlation function is then : $$\textbf{R}(\tau) = \overline{u^s(t)u^s(t+\tau)} = 2\sum_{n=1}^{N}\hat{u}^{n2} \cos(\omega_n\tau)$$

$$\omega_n$$ is a random variable defined as N($$\lambda \overline{\omega_n},\lambda' \overline{\omega_n})$$ (Normal distribution). The amplitude $$\hat{u}^n$$ only depend on the mode. Now I would like to know the results if $$N$$ tend to infinity. With the central limit theorem, one can be sure that it will tend to gaussian but I can't figure out the mean and standard deviation value of the gaussien. Any suggestion?