How to calculate the expected value of complex-valued random variable? [closed]

Question

Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the interval $[0,1]$. Then how to calculate the expected value of the following variable which is function of $\theta_1,\theta_2,\cdots, \theta_n$.

The target variable is: $|e^{j 2\pi(\theta_1-\theta_2)}+e^{j 2\pi(\theta_3-\theta_4)}+\cdots+e^{j 2\pi(\theta_{n-1}-\theta_n)}|$.
Here, $e$ is the base of the natural logarithm, $j^2=-1$ and $|\cdot|$ is the absolute value symbol.

About: This is not a homework problem. I met it when I try to calculate the expectation of $|g\star g|$, where $\star$ is the correlation operator and $|\cdot|$ represents element-wise absolute. $g$ ($g_{i,j}=e^{j2\pi\theta_{i,j}}$) is an $N\times N$ complex-valued random matrix, where $\theta_{i,j}$ are i.i.d. real random variables,

Answer 1

I am using $i=\sqrt{-1}$ instead of $j.$

Use induction, noting the differences $\theta_l-\theta_k$ are themselves uniformly distributed, so might as well just use a single $\theta_k$ in each term. For a single complex exponential the expectation is $M_1=1$.

Then find $$M_2=\mathbb{E}\left[|1+\exp(2\pi i \theta)|\right]=\mathbb{E} \left[\sqrt{(1+\cos(2\pi\theta))^2+\sin^2(2\pi\theta )}\right]$$ by integration. Whatever this mean is, and it will have a $\sin()\times\cos()$ type of term once you expand, you then add the new complex exponential term to it. So, $$M_3=\mathbb{E} \left[\sqrt{(M_2+\cos(2\pi\theta))^2+\sin^2(2\pi\theta )}\right]$$ and the integral is essentially the same.