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Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

Part of: Computational aspects Integral transforms, operational calculus Elementary classical functions

Published online by Cambridge University Press:  14 July 2016

A. G. Rossberg
A. G. Rossberg*
Affiliation:
IIASA
*
∗Postal address: Evolution and Ecology Program, International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, 2361 Laxenburg, Austria. Email address: raxel@rossberg.net
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Abstract

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It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

Keywords

Moment generating functionLaplace transformtransform inversionheavy tail

MSC classification

Secondary: 44A10: Laplace transform 33F05: Numerical approximation and evaluation 44A35: Convolution 33B15: Gamma, beta and polygamma functions
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 531 - 541
Copyright
Copyright © Applied Probability Trust 2008 

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