Editing Log-normal distribution (section)

=== Characteristic function and moment generating function ===

All moments of the log-normal distribution exist and

<math display="block">\operatorname{E}[X^n] = e^{n\mu+n^2\sigma^2/2}</math>

This can be derived by letting <math display="inline">z = \tfrac{\ln x - \mu}{\sigma} - n \sigma</math> within the integral. However, the log-normal distribution is not determined by its moments.<ref name="Heyde">{{Citation | last = Heyde | first = CC. | title = On a Property of the Lognormal Distribution | work = Journal of the Royal Statistical Society, Series B | date = 2010 | volume = 25 | issue = 2 | pages = 392–393 | doi = 10.1007/978-1-4419-5823-5_6 | isbn = 978-1-4419-5822-8 | doi-access = free}}</ref> This implies that it cannot have a defined moment generating function in a neighborhood of zero.<ref>{{Cite book | last = Billingsley | first = Patrick | url = https://www.worldcat.org/oclc/780289503 | title = Probability and Measure | date = 2012 | publisher = Wiley | isbn = 978-1-118-12237-2 | edition = Anniversary | location = Hoboken, N.J. | pages = 415 | oclc = 780289503}}</ref> Indeed, the expected value <math>\operatorname{E}[e^{t X}]</math> is not defined for any positive value of the argument <math>t</math>, since the defining integral diverges.

The [[characteristic function (probability theory)|characteristic function]] <math>\operatorname{E}[e^{i t X}]</math> is defined for real values of {{mvar|t}}, but is not defined for any complex value of {{mvar|t}} that has a negative imaginary part, and hence the characteristic function is not [[Analytic function|analytic]] at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.<ref name="Holgate">{{Cite journal | last = Holgate | first = P. | year = 1989 | title = The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989 | journal = Communications in Statistics – Theory and Methods | volume = 18 | issue = 12 | pages = 4539–4548 | doi = 10.1080/03610928908830173}}</ref> In particular, its Taylor [[formal series]] diverges:

<math display="block">\sum_{n=0}^\infty \frac{{\left(it\right)}^n}{n!} e^{n\mu + n^2\sigma^2/2}</math>

However, a number of alternative [[divergent series]] representations have been obtained.<ref name="Holgate" /><ref name="Barakat">{{Cite journal | last = Barakat | first = R. | year = 1976 | title = Sums of independent lognormally distributed random variables | journal = Journal of the Optical Society of America | volume = 66 | issue = 3 | pages = 211–216 | bibcode = 1976JOSA...66..211B | doi = 10.1364/JOSA.66.000211}}</ref><ref name="Barouch">{{Cite journal | last1 = Barouch | first1 = E. | last2 = Kaufman | first2 = GM. | last3 = Glasser | first3 = ML. | year = 1986 | title = On sums of lognormal random variables | url = http://dspace.mit.edu/bitstream/handle/1721.1/48703/onsumsoflognorma00baro.pdf | journal = Studies in Applied Mathematics | volume = 75 | issue = 1 | pages = 37–55 | doi = 10.1002/sapm198675137 | hdl = 1721.1/48703 | hdl-access = free }}</ref><ref name="Leipnik">{{Cite journal | last = Leipnik | first = Roy B. | date = January 1991 | title = On Lognormal Random Variables: I – The Characteristic Function | url = https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F1563B5AD8918EF2CD51092F82EB0B73/S0334270000006901a.pdf/div-class-title-on-lognormal-random-variables-i-the-characteristic-function-div.pdf | journal = Journal of the Australian Mathematical Society, Series B | volume = 32 | issue = 3 | pages = 327–347 | doi = 10.1017/S0334270000006901 | doi-access = free }}</ref>

A closed-form formula for the characteristic function <math>\varphi(t)</math> with <math>t</math> in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by<ref name="Asmussen">S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution",
[https://link.springer.com/article/10.1007/s11009-014-9430-7 Methodology and Computing in Applied Probability 18 (2), 441-458.]
[http://data.imf.au.dk/publications/thiele/2013/math-thiele-2013-06.pdf Thiele report 6 (13).]</ref>

<math display="block">\varphi(t) \approx \frac{\exp\left(-\frac{W^2(-it\sigma^2e^\mu) + 2W(-it\sigma^2e^\mu)}{2\sigma^2} \right)}{\sqrt{1 + W{\left(-it\sigma^2e^\mu\right)}}}</math>

where <math>W</math> is the [[Lambert W function]]. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of <math>\varphi</math>.