Skip to main content
SpringerLink
Log in

On the Laplace Transform of the Lognormal Distribution

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation \(\widetilde {\mathcal {L}}(\theta )\) of the Laplace transform \(\mathcal {L}(\theta )\) which is obtained via a modified version of Laplace’s method. This approximation, given in terms of the Lambert W(⋅) function, is tractable enough for applications. We prove that ~(𝜃) is asymptotically equivalent to ℒ(𝜃) as 𝜃 → ∞. We apply this result to construct a reliable Monte Carlo estimator of ℒ(𝜃) and prove it to be logarithmically efficient in the rare event sense as 𝜃 → ∞.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramowitz M, Stegun I (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Applied mathematics series. Dover Publications

  • Aitchison I, Brown JAC (1957) The lognormal distribution with special reference to its uses in economics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Asmussen S, Glynn PW (2007) Stochastic simulation: algorithms and analysis. Springer-Verlag, New York

    MATH  Google Scholar 

  • Asmussen S, Jensen JL, Rojas-Nandayapa L (2014) Exponential family techniques for the lognormal left tail. Research Report

  • Barakat R (1976) Sums of independent lognormally distributed random variables. J Opt Soc Am 66: 211–216

    Article  MathSciNet  Google Scholar 

  • Barouch E, Kaufman GM (1976) On sums of lognormal random variables. Working paper. A. P. Sloan School of Management, MIT

  • Beaulieu NC, Xie Q (2004) An optimal lognormal approximation to lognormal sum distributions. IEEE Trans Veh Technol 53(2):479–489

    Article  Google Scholar 

  • Butler RW (2007) Saddlepoint approximations with applications. Cambridge University Press

  • Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5:329–359

    Article  MathSciNet  MATH  Google Scholar 

  • Crow EL, Shimizu K (1988) Lognormal distributions: theory and applications. Marcel Dekker Inc., New York

    MATH  Google Scholar 

  • de Bruijn NG (1970) Asymptotic methods in analysis. Courier Dover Publications

  • Dufresne D (2008) Sums of lognormals. In: Actuarial research conference proceedings

  • Foss S, Korshunov D, Zachary S (2011) An introduction to heavy-tailed and subexponential distributions. Springer

  • Gubner JA (2006) A new formula for lognormal characteristic functions. IEEE Trans Veh Technol 55(5):1668–1671

    Article  Google Scholar 

  • Heyde C (1963) On a property of the lognormal distribution. J Roy Stat Soc Ser B 29:392–393

    MathSciNet  MATH  Google Scholar 

  • Holgate P (1989) The lognormal characteristic function. Commun Stat-Theor Methods 18:4539–4548

    Article  MathSciNet  MATH  Google Scholar 

  • Jensen JL (1994) Saddlepoint approximations. Oxford Science Publications, Oxford

    MATH  Google Scholar 

  • Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New York

  • Leipnik RB (1991) On lognormal random variables: I. the characteristic function. J Aust Math Soc Ser B 32:327–347

    Article  MathSciNet  MATH  Google Scholar 

  • Limpert E, Stahel WA, Abbt M (2001) Log-normal distributions across the sciences: keys and clues. BioScience 51:341–352

    Article  Google Scholar 

  • Rojas-Nandayapa L (2008) Risk probabilities: asymptotics and simulation. Ph.D. thesis, Aarhus University

  • Rossberg AG (2008) Laplace transforms or probability distributions and their inversions are easy on logarithmic scales. J Appl Probab 45:531–541

    Article  MathSciNet  MATH  Google Scholar 

  • Small CG (2013) Expansions and asymptotics for statistics. Chapman & Hall/CRC

  • Tellambura C, Seranarte D (2010) Accurate computation of the MGF of the lognormal distribution and its application to sum of lognormals. IEEE Trans Commun 58:1568–1577

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aarhus University, Aarhus, Denmark

    Søren Asmussen & Jens Ledet Jensen

  2. School of Mathematics and Physics, University of Queensland, Brisbane, Australia

    Leonardo Rojas-Nandayapa

Authors
  1. Søren Asmussen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Ledet Jensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leonardo Rojas-Nandayapa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonardo Rojas-Nandayapa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Asmussen, S., Jensen, J.L. & Rojas-Nandayapa, L. On the Laplace Transform of the Lognormal Distribution. Methodol Comput Appl Probab 18, 441–458 (2016). https://doi.org/10.1007/s11009-014-9430-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-014-9430-7

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation