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    ×
  • The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Volume 32, Issue 3
  • January 1991, pp. 327-347

On lognormal random variables: I-the characteristic function

  • Roy B. Leipnik (a1)
  • DOI: http://dx.doi.org/10.1017/S0334270000006901
  • Published online: 01 February 2009
Abstract
Abstract

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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]F. H. Brownell , Pacific J. Math. 5 (1955) 484491.

[2]N. J. de Bruijn , Nederl. Akad. Wetensch Proc. Series A 56 (1953) 449458; Indagationes Math 15 (1953) 459–464.

[6]A. Fransén and S. Wrigge , Math. of Comp. 34 (1980) 553566.

[12]W. Magnus , Formulas and theorems for the special functions of mathematical physics (Springer, New York, 1966).

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