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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.

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    $\begingroup$ 1) Fourier transforms/spectral analysis in time series, 2) characteristic functions, 3) sometimes contour integration used in distribution theory, ... ? quora.com/Does-complex-analysis-have-applications-in-statistics , quora.com/Is-Complex-Analysis-relevant-to-Machine-Learning $\endgroup$ Commented Feb 24, 2019 at 20:48
  • $\begingroup$ Ah, yes! Of course regarding transforms/spectral in time series! Thanks for this! and the other suggestions too! $\endgroup$ Commented Feb 24, 2019 at 20:52
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    $\begingroup$ The QM applications don't actually use complex numbers for the statistical calculations: all probability amplitudes must be real numbers (that's axiomatic). Complex numbers are required to model interference of waves, but once that is taken care of, only their amplitudes are considered for statistical calculations. This is the thrust of the requirement that all QM operators be Hermitean. $\endgroup$
    – whuber
    Commented Feb 25, 2019 at 13:43
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    $\begingroup$ Thanks, @whuber. My knowledge of quantum mechanics is about as vast as art history: virtually 0. ;-) $\endgroup$ Commented Feb 25, 2019 at 16:27

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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.)

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    $\begingroup$ Thanks, @Ben. Tha'ts great. I'l going to leave the question open for a few days to see if I get some other responses too. Otherwise, this is a perfectly good answer. I'll come back in a couple of days to accept. $\endgroup$ Commented Feb 25, 2019 at 2:14
  • $\begingroup$ Another ide which might be mentioned is analytic function theory which crops up in math stat, for instance in unicity proofs, completeness. Maybe one van do with real analytic functions, but mostly complex is used. $\endgroup$ Commented Mar 6, 2019 at 11:54

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