How to formally define a probability distributions over complex random variables?

Question

Would that be just a probability over a bivariate real random variable, one representing the real part and another representing the imaginary part? How can I formally take moments of the complex random variable (just take moments separately and combine both moments, one of the real part and another of the imaginary part)? Also, can you show real examples where this formalization is applied?

Answer 1

Quoting from the detailed Wikipedia page on this topic,

Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts.

and

A complex random variable ${\displaystyle Z}$ on the probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ is a function $${\displaystyle Z\colon \Omega \rightarrow \mathbb {C} }$$ such that both its real part $ {\displaystyle \Re {(Z)}}$ and its imaginary part ${\displaystyle \Im {(Z)}}$ are real random variables on $(\Omega ,{\mathcal {F}},P)$.

A complex random variable $Z=X+\iota Y$ is equivalent to a bivariate real random vector $(X,Y)$ and its distribution is thus defined by the joint distribution of $(X,Y)$, e.g., through a density $f(x,y)$. Any probabilistic statement about $Z$ can equally be stated as a probabilistic statement about $(X,Y)$. The possible source of confusion about a complex random variable is that the density $f(z)$ may appear to be a complex valued function but this is not the case, $f$ is defined as$$f:\mathbb {C}\longrightarrow\mathbb {R}^+$$Another way to look at this issue is to separate the vectorial space structure of $\mathbb {C}$ from its algebraic field structure, which does not not directly impact the probability measure.