Distribution of the direction of Gaussian random variable

Question

Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on it?

What about the multivariate case? That is, I have a multivariate complex normal, and would like to understand the multivariate distribution of the phases of the individual components.

Does anything become easier if the real and imaginary parts are uncorrelated? (In the multivariate case, uncorrelated for each component.)

Answer 1

In the 2D case, you can write $X=AZ$, where $A$ is a $2\times2$ nonsingular real matrix, $Z:=[Z_1,Z_2]^T$, and the $Z_j$'s are iid standard normal. You want to find $$p:=P(n_1\cdot X>0,\;n_2\cdot X>0),$$ where $n_1$ and $n_2$ are unit vectors in $\mathbb R^2$ and $\cdot$ is the dot product.

By the rotational symmetry of the distribution of $Z$, you have $$p=P(m_1\cdot Z>0,\;m_2\cdot Z>0)=\frac1{2\pi}\arccos\frac{m_1\cdot m_2}{|m_1|\,|m_2|},$$ where $m_j:=A^T n_j$ and $|\cdot|$ is the Euclidean norm on $\mathbb R^2$.

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When the dimension is $>2$, the problem similarly reduces to finding the probability that a standard normal random vector is in a polyhedral cone. This is a difficult problem, admitting a certain recursive solution, which can be resolved more or less explicitly for dimensions $\le4$. See e.g. Plackett and references there, notably to Schläfli.