Hostname: page-component-5db58dd55d-l8wb7 Total loading time: 0 Render date: 2026-05-30T16:12:07.091Z Has data issue: false hasContentIssue false
  • English
  • Français

On lognormal random variables: I-the characteristic function

Published online by Cambridge University Press:  17 February 2009

Roy B. Leipnik
Roy B. Leipnik
Affiliation:
Mathematics Department, University of California at Santa Barbara, CA 93106, USA.
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.

The characteristic function of a lognormal random variable is calculated in closed form as a rapidly convergent series of Hermite functions in a logarithmic variable. The series coefficients are Nielsen numbers, defined recursively in terms of Riemann zeta functions. Divergence problems are avoided by deriving a functional differential equation, solving the equation by a de Bruijn integral transform, expanding the resulting reciprocal Gamma function kernel in a series, and then invoking a convergent termwise integration. Applications of the results and methods to the distribution of a sum of independent, not necessarily identical lognormal variables are discussed. The result is that a sum of lognormals is distributed as a sum of products of lognormal distributions. The case of two lognormal variables is outlined in some detail.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 3 , January 1991 , pp. 327 - 347
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Brownell, F. H., Pacific J. Math. 5 (1955) 484–491.Google Scholar
[2]de Bruijn, N. J., Nederl. Akad. Wetensch Proc. Series A 56 (1953) 449–458; Indagationes Math 15 (1953) 459–464.CrossRefGoogle Scholar
[3]Carleman, T., Les fonctions quasi-analytiques (Gauthier-Villars, Paris, 1926).Google Scholar
[4]Crow, E. L. and Shimizu, K., Lognormal distributions: Theory and applications (Dekker, New York, 1988).Google Scholar
[5]Erdelyi, A., et al, Tables of integral transforms (Vol. 1, McGraw-Hill, New York, 1954).Google Scholar
[6]Fransén, A. and Wrigge, S., Math. of Comp. 34 (1980) 553–566.CrossRefGoogle Scholar
[7]Gradshtein, I. and Ryzhik, I., Tables of integrals, series and products (2nd transl. ed., Academic Press, New York, 1980).Google Scholar
[8]Holgate, P., “The lognormal characteristic function”, Comm Stat. Theory (In Press).Google Scholar
[9]Krein, M., C. R. (Doklady) Acad. Sci. URSS (NS) 40 (1945) 306–309.Google Scholar
[10]Leipnik, R. B., “The lognormal distribution and strong nonuniqueness of the moment problem”, J. Prob. Appl. (1981) 863–865.Google Scholar
[11]Levin, B. J., Distribution of zeros of entire functions, v5., Translations of Mathematical Monographs, AMS (1964).Google Scholar
[12]Magnus, W. et al. , Formulas and theorems for the special functions of mathematical physics (Springer, New York, 1966).CrossRefGoogle Scholar
[13]Nielsen, N., Handbuch der Theorie der Gammafunktion (Teubner, Leipzig, 1906).Google Scholar
[14]Pinney, E., Ordinary difference-differential equations (University of California, Berkeley, 1958).Google Scholar
[15]Sansone, G., Orthogonal functions (Interscience, New York, 1959).Google Scholar
[16]Titchmarsh, E. G., The theory of fourier integrals (Oxford University Press, Oxford, 1948).Google Scholar