The sum I am looking for the following sum when $M \; -> \; \infty$.

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

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 $+\pi$.

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