I have the following function.
$$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$
Here, $U_m$ samples are random numbers coming from a Gaussian distribution
$$U_m \sim \mathcal{N}(\mu_u, \sigma_u)$$
and the $\beta_m$ values come randomly from a uniform distribution.
$$ \beta_m \sim \mathcal{U}[0, 2\pi] $$
If say I have $N$ samples of $U_m$ and $\beta_m$ for each $k$ in the series, I found that $x(k)$ will have a Gaussian distribution for a large number of $k$ samples (realizations) with a mean $\mu_x = 0$ and a standard deviation of $\sigma_x = \sqrt{N}$ (This comes from central limit theorem). Let's define the number of $k$ samples (realizations) as $M$.
Although through simulations I see that the value of $\sigma_x$ for both real and imaginary parts of $x(k)$ converge for a large $M$ with a value $\sigma_{x, converged} = \sqrt{N/2}$.
I wan to know if I can find an expression for $M$ as a function of $\mu_u, \sigma_u$ where convergence occurs for $\mu_x$ and $\sigma_x$. It can be based on a precision input $\epsilon$ like, $\epsilon = 10^{-4}$.
So, basically I want something like this:
$$ M|_{\mu_x, \sigma_x} = f(\epsilon, \mu_u, \sigma_u, N) $$