Complex Random Variables

Rarely have casualty actuaries needed, much less wanted, to work with complex numbers. One readily could wisecrack about imaginary dollars and creative accounting. However, complex numbers are well established in mathematics; they even provided the impetus for abstract algebra. Moreover, they are essential in several scientific fields, most notably in electromagnetism and quantum mechanics, the two fields to which most of the sparse material about complex random variables is tied. This paper will introduce complex random variables to an actuarial audience, arguing that complex random variables will eventually prove useful in the field of actuarial science. First, it will describe the two ways in which statistical work with complex numbers differs from that with real numbers, viz., in transjugation versus transposition and in rank versus dimension. Next, it will introduce the mean and the variance of the complex random vector, and derive the distribution function of the standard complex normal random vector. Then it will derive the general distribution of the complex normal multivariate and discuss the behavior and moments of complex lognormal variables, a limiting case of which is the unit-circle random variable W=ei for real ? uniformly distributed. Finally, it will suggest several foreseeable actuarial applications of the preceding theory, especially its application to linear statistical modeling. Though the paper will be algebraically intense, it will require little knowledge of complex-function theory. But some of that theory, viz., Cauchy’s theorem and analytic continuation, will arise in an appendix on the complex moment generating function of a normal random multivariate.

Keywords: Complex numbers, matrices, and random vectors; augmented variance; lognormal and unit-circle distributions; determinism; Cauchy-Riemann; analytic continuation