Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform
 Richard L. Warr
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First passage distributions of semiMarkov processes are of interest in fields such as reliability, survival analysis, and many others. Finding or computing first passage distributions is, in general, quite challenging. We take the approach of using characteristic functions (or Fourier transforms) and inverting them to numerically calculate the first passage distribution. Numerical inversion of characteristic functions can be unstable for a general probability measure. However, we show they can be quickly and accurately calculated using the inverse discrete Fourier transform for lattice distributions. Using the fast Fourier transform algorithm these computations can be extremely fast. In addition to the speed of this approach, we are able to prove a few useful bounds for the numerical inversion error of the characteristic functions. These error bounds rely on the existence of a first or second moment of the distribution, or on an eventual monotonicity condition. We demonstrate these techniques with two examples.
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 Title
 Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform
 Journal

Methodology and Computing in Applied Probability
Volume 16, Issue 4 , pp 10251038
 Cover Date
 20141201
 DOI
 10.1007/s1100901393663
 Print ISSN
 13875841
 Online ISSN
 15737713
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Characteristic function
 First passage distribution
 Fast Fourier transform
 SemiMarkov process
 30E10
 60G10
 60K15
 Industry Sectors
 Authors

 Richard L. Warr ^{(1)}
 Author Affiliations

 1. Air Force Institute of Technology, Dayton, OH, USA