Is it possible to write this lognormal PDF in terms of the Meijer-G function?
$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{(-10\log_{10}(y) - \mu)^2}{2 \sigma^2}\right)$$
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asked Oct 9, 2021 at 12:07
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1$\begingroup$ no idea why you would want to do this, but it's certainly possible: $$f_Y(y)=5(\sigma y \ln 10)^{-1} \sqrt{\frac{2}{\pi }} \exp \left(-(2 \sigma^2)^{-1}\left((10/\ln 10) \left(\text{MeijerG}\left[\{\{1,1\},\{\}\},\left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right),y\right]-\text{MeijerG}\left[\{\{1,1\},\{\}\},\left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right),y,-1\right]\right)+\mu\right)^2\right)$$ $\endgroup$– Carlo BeenakkerCommented Oct 9, 2021 at 12:14
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$\begingroup$ @CarloBeenakker, thanks for your reply. I've got a summation of L lognormal RVs and I'd like to find the resulting PDF or an approximation. $\endgroup$– Felipe Augusto de FigueiredoCommented Oct 9, 2021 at 12:36
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1$\begingroup$ @CarloBeenakker: This looks like the very same expansion with logarithm written in an awkward way, using $$G_{22}^{12}(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\,|\,x)=\log(1+x)$$ and $$G_{22}^{21}(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\,|\,x)=\log(1+x^{-1}),$$ as well as Mathematica's convention for generalised Meijer G-function. Not really useful for anything one might want to do with the representation in terms of Meijer G-function (which is often used for integration and integral transforms, I suppose). $\endgroup$– Mateusz KwaśnickiCommented Oct 9, 2021 at 20:25
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1$\begingroup$ @FelipeAugustodeFigueiredo: I am no expert here, but I do not think this is possible, at least not in any useful way. If I am not mistaken, Meijer G-function cannot decay that fast near zero (or infinity) — but once again, I may be completely wrong here, my experience with Meijer G-function is very limited. You may like to check Prudnikov's three-volume book with its enormous table of expansions of different functions in terms of the Meijer G-function. $\endgroup$– Mateusz KwaśnickiCommented Oct 9, 2021 at 20:35
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1$\begingroup$ Here's a reference to the book: Prudnikov, A.P., Brychkov, Yu A., Marichev, O.I.: Integrals and Series, Vol. 3: More Special Functions. Gordon and Breach Science Publishers, New York (1990). $\endgroup$– Mateusz KwaśnickiCommented Oct 9, 2021 at 20:36
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