Use of Complex Numbers in Statistics I was asked recently if complex numbers were used in Statistics by a friend of mine who is an electrical engineer.  Besides statistical applications in other fields (e.g. quantum mechanics) and besides some  characteristic functions, I couldn't really identity many any off the top of my head.  I'm curious if anyone else can identity uses of complex numbers in statistics, outside of applications in other fields.  
 A: There are two broad classes of use of complex numbers in statistics, one being when the underlying problem uses complex numbers (leading to complex random variables), and the other being when tools using complex numbers are used to describe statistical problems involving only real random variables.  (I will leave aside theories of probability that use complex probabilities - these are pretty crazy and I have never been able to see the use in them.)
Problems involving complex random variables: Complex numbers arise in statistics whenever you are dealing with an underlying problem where the random variables of interest are complex numbers themselves (i.e., complex random variables).  These applications tend to arise in the context of engineering and physics problems where complex numbers are used to describe some phenomena of interest, and we wish to add randomness to the description of that phenomena.  In particular, they often come up in the context of dealing with circular motion that is described by complex numbers, or in electrical circuits.  There is already a substantial statistical literature on this field, including results for complex versions of normal random variables, etc. (for an overview, see e.g., Eriksson et al 2009, Eriksson et al 2010).  (Note: Since your friend is an electrical engineer, it might be worth pointing him to various works published in IEEE that deal with complex random variables.  These are often used in electrical engineering work when the analyst wishes to add randomness to some aspect of an underlying electrical problem that uses complex numbers.)
Another common example of this kind of situation is whenever you have polynomials with randomly generated real coefficients, where the distribution of the coefficients is continuous.  In this case, even though the initial random variables are real, this gives rise to complex roots of the polynomial, so when you write the polynomial in its factorised form, this involves complex random variables (see e.g., Shepp and Vanderbei 1995, Ibragimov and Zeitouni 1997, Kabluchko and Zaporozhets 2014).  This is a simple example where random objects involving real numbers give rise to complex random variables.
Complex tools for dealing with real random variables: The most common set of statistical tools that deal with real random variables, but use complex numbers, are tools that are applications of the Fourier transform to various statistical problems.  This includes the characteristic function used to describe a distribution in Fourier-space,  frequency-space periodograms used for identifying signal frequencies in time-series analysis, and the various spectral densities in time-series analysis.  These are all examples where a standard mathematical tool using complex numbers is applied in a probability problem where the underlying random variables are real numbers.  These methods are often used in time-series analysis, but also crop up sometimes when dealing with tricky probability problems involving convolutions.  (In fact, the simplest proof of the central limit theorem uses the characteristic function of the normal distribution, so it involves the use of complex numbers.)
