1 The Discrete Fourier Transform Leigh J. Halliwell, FCAS, MAAA Brian Fannin, ACAS, MAAA CASE Spring Meeting Middle Tennessee State University March 22, 2016

2 Antitrust Notice The Casualty Actuarial Society is committed to adhering strictly to the letter and spirit of the antitrust laws. Seminars conducted under the auspices of the CAS are designed solely to provide a forum for the expression of various points of view on topics described in the programs or agendas for such meetings. Under no circumstances shall CAS seminars be used as a means for competing companies or firms to reach any understanding – expressed or implied – that restricts competition or in any way impairs the ability of members to exercise independent business judgment regarding matters affecting competition. It is the responsibility of all seminar participants to be aware of antitrust regulations, to prevent any written or verbal discussions that appear to violate these laws, and to adhere in every respect to the CAS antitrust compliance policy.

3 Outline I.Complex Numbers II.Roots-of-Unity Random Variables III.Collective-Risk Model IV. Examples

4 I. Complex Numbers Basic Arithmetic Complex Plane versus Real Line Wessel(1979)-Argand(1806) diagram [Kramer, 73] No trichotomy between complex numbers

5 I. Complex Numbers (Cont’d) Completeness of Complex Numbers Fundamental Theorem of Algebra [Kramer, 71] Actuaries should examine the meaning of their formulas with complex numbers. Or are they just “imaginary” numbers? Conjugate Theorems

6 I. Complex Numbers (Cont’d) The exponential function Absolutely convergent, since k! outgrows z k.

7 I. Complex Numbers (Cont’d) The real exponential is 1-to-1 (increasing) every unit is e more than before. “Real” exponential growth The wheat and chess problem What is “imaginary” exponential growth? The distance of e z from O is invariant to the imaginary part of z We’re almost to Euler’s famous formula (see next)

8 I. Complex Numbers (Cont’d) Velocity here is proportional to position But multiplication by i is counterclockwise rotation by 90° So f(t) is circular motion; must repeat every 2 π radians “Will it Go Round in Circles?” Billy Preston 1972 Euler’s Formula: “Imaginary” growth is circular, with period 2πi The basis of polar coordinates:

I. Complex Numbers (Cont’d) Never again need you forget your trig identities! Physical or Actuarial Application?? As, becomes uniform circular Infinite variance can be “domesticated” exponentially Integral powers of are dense but unordered on Κ – chaos

10 I. Complex Numbers (Cont’d) What is legitimate exponentiation, or z s ? Use logs: Uniquely defined only for integral values of s. like a double-slit experiment that recombines before we can see it real exponential allows for well defined (x>0) y, but not extendable to C.

11 I. Complex Numbers (Cont’d) The Circle Group (multiplicative): closed under multiplication a) associative b) identity element c) inverse: The “n th Roots of Unity” Group primitive root ω n distinct numbers

12 I. Complex Numbers (Cont’d) Excel Graphs of Ω n Findings n th roots of unity sum to zero for n > 1 The k th powers of Ω n equal Ω m, where m = n/GCF(k, n) e.g., squares of tenth roots are the fifth roots of unity e.g., outer join of second and third roots = sixth roots e.g., outer join of second and fourth roots = fourth roots e.g., if n is prime, k th powers of Ω n equal Ω n for 0 < k < n Ω n groups are useful in number theory (prime factorization)

13 II. Roots-of-Unity Random Variables The Ω n RV: Because, only n distinct moments:

14 II. Roots-of-Unity Random Variables (Cont’d) Ω is the (nxn) discrete Fourier transform symmetric: invertible: One-to-one matching between probabilities and moments. Ω  1 is the inverse Fourier transform

15 II. Roots-of-Unity Random Variables (Cont’d) If Z 1 and Z 2 are independent Ω n RVs, then: Isomorphism between and : Circles fundamental to lines, for lines are circles of infinite radius.

16 III. Collective-Risk Model Severities X and claim count N are all independent: If N ~ Poisson( λ ): Exact probabilities under arithmetic modulo n What is the mean of an exponential RV wrapped around limit l ?

17 IV. Examples 1. Binomial probabilities to overflow 2. Klugman example 3. Non-probabilistic DFT uses a. Binomial coefficients b. Combining AQ pattern in AY pattern

18 References Clark, Allan, Elements of Abstract Algebra, New York: Dover, Introduction: theory of equations as the start of modern algebra - Chapter 2: group theory, circle group, cosets, homo- and isomorphisms Halliwell, L., “The Discrete Fourier Transform and Cyclical Overflow,” Variance, 8:1, 2014, 73-79, “, “Complex Random Variables,” CAS E-Form, Fall 2015, Kramer, Edna E., The Nature and Growth of Modern Mathematics, Princeton University Press, Ch 4 (70-82): complex numbers, Hamilton, quaternions - Ch 28: Grassman, Noether, twentieth-century algebra and physics Klugman, S. A., H. H. Panjer, and G. E. Willmot, Loss Models: From Data to Decisions, New York: Wiley, §4.7.1: fast Fourier transform