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The lognormal characteristic function. (English) Zbl 0721.60012
Summary: A number of different ways are examined of representing the characteristic function \(\phi\) (t) of the lognormal distribution, which cannot be expanded in a Taylor series based on the moments. In § 2 the use of a finite Taylor series is examined. A method of summing the divergent formal expansion is discussed in § 3. In § 4 the fact that \(\phi\) (t) is a boundary analytic function is exploited. Asymptotic approximation of the integral defining \(\phi\) (t) is studied in § 5. Each approach produces some interesting information about the distribution.

MSC:
60E10 Characteristic functions; other transforms
62E20 Asymptotic distribution theory in statistics
62E10 Characterization and structure theory of statistical distributions
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References:
[1] Aitchison J., The lognormal distribution (1957) · Zbl 0081.14303
[2] Crow E.L., Lognormal distributions (1988) · Zbl 0644.62014
[3] De Bruin N.G., Asymptotic methods in analysis (1961)
[4] Hardy G.H., Divergent Series (1949) · Zbl 0032.05801
[5] Heyde C.C., J. Roy, Statist Soc. B 25 pp 392– (1963)
[6] DOI: 10.1137/1126095 · Zbl 0488.60024 · doi:10.1137/1126095
[7] Lukacs E., Characteristic functions (1970) · Zbl 0201.20404
[8] Thorin O., Scand. Act J pp 121– (1977)
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