Talk:Laplace–Stieltjes transform

At the Probability distribution sections, the convolution is NOT done over the CDF-s but over the PDF-s, as it is correctly written in the LTS integral. Am I right? In every other source I have found that the convolution should be done on the PDF-s to get the PDF of the sum of the variables. e.g.https://en.wikipedia.org/wiki/Convolution_of_probability_distributions

Does the convolution theorem *actually* hold for this?

Assume that $f,g:\mathbb {R} \to \mathbb {R}$ are both differentiable, with their support only being in the positive real numbers. Then

${\begin{aligned}{\mathcal {L}}^{*}(f*g)&={\mathcal {L}}((f*g)')={\mathcal {L}}\left({\frac {d}{dx}}\int _{0}^{\infty }f(t)g(x-t)\,dt\right)=-{\mathcal {L}}\left(\int _{0}^{\infty }f(t)g'(x-t)\,dt\right)\\&=-{\mathcal {L}}(f*g')=-{\mathcal {L}}(f){\mathcal {L}}(g')=-{\mathcal {L}}(f){\mathcal {L}}^{*}(g),\end{aligned}}$

where I used differentiation under the integral sign. This seems to suggest the article is wrong about this, no? --Svennik (talk) 19:01, 15 February 2024 (UTC)