Journal of Applied Probability

A technique for computing the PDFs and CDFs of nonnegative infinitely divisible random variables

Mark S. Veillette and Murad S. Taqqu

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We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform EeX = e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.

Article information

J. Appl. Probab., Volume 48, Number 1 (2011), 217-237.

First available in Project Euclid: 15 March 2011

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Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 65C50: Other computational problems in probability 60-08: Computational methods (not classified at a more specific level) [See also 65C50]
Secondary: 60-04: Explicit machine computation and programs (not the theory of computation or programming)

Infinitely divisible distribution Post-Widder formula stable distribution stochastic integration


Veillette, Mark S.; Taqqu, Murad S. A technique for computing the PDFs and CDFs of nonnegative infinitely divisible random variables. J. Appl. Probab. 48 (2011), no. 1, 217--237. doi:10.1239/jap/1300198146.

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