The expectation of absolute value of the sum of n i.i.d. random variables

Question

Let $\varphi_i$ be a Gaussian random variable such that $$\varphi_i \sim N(0,\sigma^2), \quad i = 1,2,\ldots,n.$$ What's the expectation: $$E\left(\left | \sum_{i=1}^n e^{j \varphi_i} \right |\right) $$ where $\left | \cdot \right |$ is the absolute value operation and $j = \sqrt{-1}$.

Answer 1

The expectation you are interested in is equivalent to

$$E\left[\sqrt{n+2\sum_{j=1}^{n-1}\sum_{k=j+1}^n (\cos \varphi_j \cos \varphi_k+\sin\varphi_j \sin\varphi_k)}\right]$$

I don't believe there is a nice closed-form solution for that expectation. So I think you'll need to perform simulations or numerically integrate to get solutions.

If you were interested in

$$E\left[n+2\sum_{j=1}^{n-1}\sum_{k=j+1}^n (\cos \varphi_j \cos \varphi_k+\sin\varphi_j \sin\varphi_k)\right]$$

then there is a closed-form solution:

$$n+n(n-1)e^{-\sigma^2}$$