The sum I am looking for is the following sum as $M \to \infty$:

$$ L(\omega) = \sum_{m = 1}^{M} \frac{\sin\left( N \frac{\omega_m - \omega}{2} \right)}{\sin\left( \frac{\omega_m - \omega}{2} \right)} \cos\left(N \frac{\omega_m - \omega}{2} + \beta_m \right) $$

where

- $\omega_m$ is a random number from a Gaussian distribution having the parameters (mean $\mu$ and variance $\sigma^2$). $$ \omega_m \sim \mathcal{N}(\mu, \sigma^2) $$
- the $\beta_m$ are random numbers drawn from a uniform distribution from $-\pi$ to $+\pi$,

$$ \beta_m \sim \mathcal{U}[-\pi, +\pi] $$