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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?

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