Expectation of a powered complex circular gaussian process

Question

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$.

The question is: $\mathbb{E}\{\textbf{x}^n\} = ~?$ $\left(n\in\mathbb{N}^{*+}\right)$

PS: Through some Matlab simulations, the result seems to be equal to $0$ for all $n$.

Answer 1

Since $x=R\mathrm e^{\mathrm i\Theta}$ where $R$ and $\Theta$ are independent, $E(x^n)=E(R^n)E(\mathrm e^{n\mathrm i\Theta})$. Furthermore, $E(\mathrm e^{n\mathrm i\Theta})=0$ for every integer $n$ because $\Theta$ is uniform on $(0,2\pi)$, hence $E(x^n)=0$.