Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T04:21:21.099Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Integral transforms, operational calculus Computational aspects 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: r[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

[1] Abate, J. and Valkó, P. P. (2004). Multi-precision Laplace transform inversion. Internat. J. Numerical Meth. Eng. 60, 979–993.CrossRefGoogle Scholar
[2] Abate, J. and Whitt, W. (1999). Computing Laplace transforms for numerical inversion via continued fractions. INFORMS J. Comput. 11, 394–405.CrossRefGoogle Scholar
[3] Abate, J. and Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS J. Comput. 18, 408–421.CrossRefGoogle Scholar
[4] Abu-Dayya, A. and Beaulieu, N. C. (1994). Outage probabilities in the presence of correlated lognormal interference. IEEE Trans. Veh. Technol. 43, 164–173.CrossRefGoogle Scholar
[5] Asmussen, S. and Kroese, D. P. (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545–558.CrossRefGoogle Scholar
[6] Beaulieu, N. C. and Rajwani, F. (2004). Highly accurate simple closed-form approximations to lognormal sum distributions and densities. IEEE Commun. Lett. 8, 709–711.CrossRefGoogle Scholar
[7] Beaulieu, N. C. and Xie, Q. (2004). An optimal lognormal approximation to lognormal sum distributions. IEEE Trans. Veh. Technol. 53, 479–489.CrossRefGoogle Scholar
[8] Beaulieu, N. C., Abu-Dayya, A. A. and McLane, P. J. (1995). Estimating the distribution of a sum of independent lognormal random variables. IEEE Trans. Commun. 43, 2869–2873.CrossRefGoogle Scholar
[9] Den Iseger, P. (2006). Numerical transform inversion using Gaussian quadrature. Prob. Eng. Inf. Sci. 20, 1–44.CrossRefGoogle Scholar
[10] Dubner, H. and Abate, J. (1968). Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J. Assoc. Comput. Mach. 15, 115–123.CrossRefGoogle Scholar
[11] Duffield, N. G. and Whitt, W. (2000). Network design and control using on/off and multilevel source traffic models with heavy-tailed distributions. In Self-Similar Network Traffic and Performance Evaluation, eds Park, K. and Willinger, W., John Wiley, New York, pp. 421–445.CrossRefGoogle Scholar
[12] Dufresne, D. (2004). The log-normal approximation in financial and other computations. Adv. Appl. Prob. 36, 747–773.CrossRefGoogle Scholar
[13] Frisch, U. and Sornette, D. (1997). Extreme deviations and applications. J. Physique II 7, 1155–1171.Google Scholar
[14] Gaver, D. P. (1966). Observing stochastic processes and approximate transform inversion. Operat. Res. 14, 444–459.CrossRefGoogle Scholar
[15] Gradshtein, I. and Ryzhik, I. (1980). Tables of Integrals, Series and Products, 2nd edn. Academic Press, New York.Google Scholar
[16] Gross, D. and Harris, C. M. (1998). Fundamentals of Queueing Theory, 3rd edn. John Wiley, New York.Google Scholar
[17] Leipnik, R. R. (1991). On lognormal random variables. I. The characteristic function. J. Austral. Math. Soc. Ser. B 32, 327–347.CrossRefGoogle Scholar
[18] Lopez-Fidalgo, J. and Sanchez, G. (2005). Statistical criteria to establish bioassay programs. Health Phys. 89, 333–338.CrossRefGoogle ScholarPubMed
[19] Milevsky, M. A. and Posner, S. E. (1998). Asian options, the sum of lognormals, and the reciprocal gamma distribution. J. Financial Quant. Anal. 33, 409–422.CrossRefGoogle Scholar
[20] Nielsen, N. (1906). Handbuch der Theorie der Gammafunktion. Teubner, Leipzig.Google Scholar
[21] Romeo, M., Costa, V. D. and Bardou, F. (2003). Broad distribution effects in sums of lognormal random variables. Europ. Phys. J. B 32, 513–525.CrossRefGoogle Scholar
[22] Rossberg, A. G., Amemiya, T. and Itoh, K. (2008). Accurate and fast approximations of moment-generating functions and their inversion for log-normal and similar distributions. Submitted.Google Scholar
[23] Shortle, J. F. et al. (2004). An algorithm to compute the waiting time distribution for the {M/G/1} queue. INFORMS J. Comput. 16, 152–161.CrossRefGoogle Scholar
[24] Slimane, S. B. (2001). Bounds on the distribution of a sum of independent lognormal random variables. IEEE Trans. Commun. 49, 975–978.CrossRefGoogle Scholar
[25] Stehfest, H. (1970). Algorithm 368: numerical inversion of Laplace transforms. Commun. ACM 13, 47–49.CrossRefGoogle Scholar
[26] Talbot, A. (1979). The accurate inversion of Laplace transforms. J. Inst. Maths. Appl. 23, 97–120.CrossRefGoogle Scholar
[27] Valkó, P. and Abate, J. (2004). Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Comput. Math. Appl. 48, 629–636.CrossRefGoogle Scholar
[28] Valkó, P. P. and Vojta, B. L. (2003). The list. http://www.pe.tamu.edu/valko/Nil/LapLit.pdf.Google Scholar
[29] Vallade, M. and Houchmandzadeh, B. (2003). Analytical solution of a neutral model of biodiversity. Phys. Rev. E 68, 061902.CrossRefGoogle ScholarPubMed