The Discrete Fourier Transform and Cyclical Overflow
By Leigh Joseph Halliwell
More casualty actuaries would employ the discrete Fourier transform (DFT) if they understood it better. In addition to the many fine papers on the DFT, this paper might be regarded as just one more introduction. However, the topic uniquely explained herein is how the DFT treats the probability of amounts that overflow its upper bound, a topic that others either have not noticed or have deemed of little importance. The cyclical overflow originates in the modular arithmetic whereby the DFT evaluates characteristic functions. To understand this is to attain a deeper understanding of the DFT, which may lead to its wider use.
KEYWORDS Collective-risk model, discrete Fourier transform, characteristic function, roots of unity