I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?
All the best,
Pierre-O.
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$\begingroup$ Please note that the orthogonal polynomials for the lognormal distribution are not dense in $L_2$ equipped with the measure induced by the lognormal (see the article by Ernst et. al. dx.doi.org/10.1051/m2an/2011045). $\endgroup$– Elmar ZanderCommented Jun 10, 2016 at 9:23
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$\begingroup$ Yes, I know see proposition 1.1 in pierre-olivier.goffard.me/Publications/… $\endgroup$– LaGauffreBZHCommented Jun 11, 2016 at 13:01
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$\begingroup$ Maybe less technical than what is proposed in the article you make reference to. I will definitely have a look at it even though I am not sure to understand. Thanks for the comment! $\endgroup$– LaGauffreBZHCommented Jun 11, 2016 at 13:04
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$\begingroup$ Interesting article. I've never seen the approach in approximating density ratios. We only do expansions of random variables in orthogonal polynomials and I frequently have to warn people (students/engineers) not to use lognormals as basis random variables in their expansions. So maybe it's good that the comment is there to warn other people. $\endgroup$– Elmar ZanderCommented Jun 15, 2016 at 10:52
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$\begingroup$ Expansion of random variables? Is it related to Malliavin calculus? $\endgroup$– LaGauffreBZHCommented Jun 16, 2016 at 12:13
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Orthogonal polynomials with respect to the lognormal distribution go by the name of Stieltjes-Wigert polynomials. Two recent studies of their properties:
answered Aug 26, 2015 at 15:51
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$\begingroup$ Thank you, let see if this factor 1/x in the lognormal pdf is not an issue. Cheers. $\endgroup$ Commented Aug 27, 2015 at 7:47