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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.)

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    $\begingroup$ This problem arises immediately in MIMO radar direction estimation. See, for example, MUSIC $\endgroup$ May 26, 2020 at 16:56

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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 $\mathbb R^2$.


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.

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  • $\begingroup$ I don't necessarily want to limit myself to the case where $\mathbb{E}(X) = 0$, and this method seems limited to that case, correct? $\endgroup$ May 26, 2020 at 19:41
  • $\begingroup$ @ElenaYudovina : Yes, this approach works only for $EX=0$. However, the problem is hardly just with this approach. Even for the 2D case, Mathematica cannot find a closed-form expression for general $EX\ne0$ -- I think it does not exist. You can try to use the dimension reduction as in the linked paper by Plackett, but I think an ordinary integral which cannot be taken in closed form will then still remain. Alas, not many integrals can be taken in closed form! $\endgroup$ May 26, 2020 at 21:22

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