random process - What is the distribution of it?

If $\theta$ is uniformly distributed in $(0, 2\pi),$ then what is the distribution of $e^{i\theta},$ where $i = \sqrt{-1}?$ And what are the statistical properties of $\left[e^{i0\theta}\, e^{i1\theta}\, e^{i2\theta}\dots\, e^{i(N-1)\theta}\right],$ where $\theta$ is a single observation and $N$ some integer?