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 $+\pi$,


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



Is it possible to sum this analytically in any way?
Since the average over $\beta$ will give a vanishing expectation value of $L$, let me omit it for now and set $\beta=0$. I will also simplify the question by setting $\omega=\mu=0$ and $\sigma=1$. Then the expectation value of $L$ has a compact expression
$$\mathbb{E}[L(0)]=M\;\int\_{-\infty}^\infty \frac{dx}{\sqrt{2\pi}}e^{-x^2/2} \frac{\sin\left( N x/2 \right)}{\sin\left( x/2\right)} \cos\left(N x/2 \right)=Me^{-\frac{1}{8} (2 N-1)^2} \sum \_{j=1}^N e^{-\frac{1}{2} (j-1) (j-2 N)}.$$


