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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.
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Abstract

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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.

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

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