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The following situation arises frequently in probability. Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the convolution of the pdfs of $X$ and $Y$: $$f_Z(z)=\int_{x+y=z}f(x)f(y) dxdy$$ Because a Fourier transform converts the convolution to a pointwise product, we can often analyze $Z$ pretty well.

I am interested, instead, in the product $W=XY$. In my particular case, $X$ and $Y$ (and hence $W$) lie on the open unit disk in the complex plane (that is, $X,Y\in\mathbb{C}, |X|,|Y|<1$). It's simple to write down the pdf of their product: $$f_W(w)=\int_{xy=w}f(x)f(y) dxdy$$ Does this object have a name? Any special properties? Something analogous to a Fourier transform I should be looking at? (Incidentally, I'm happy to set $f_X(0)=f_Y(0)=0$ to avoid some pathologies.)

Although my particular application lies on the complex plane, one can naturally ask the same question, for example, on the reals or the unit quaternions. I'm aware of a similar operation over the integers, Dirichlet convolution, but I'm not sure how this operation helps with the continuous case.

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    $\begingroup$ Did you look into convolutions on groups? $\endgroup$ – Bjørn Kjos-Hanssen Nov 5 '14 at 17:14
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It's multiplicative convolution. What you can look at is the Mellin transform that behaves for this multiplicative convolution like the Fourier transform for additive convolution.

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Or if your random variables are nonnegative and nicely padded away from zero, you can take logorithms and come back to standard convolution.

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    $\begingroup$ That's great over the reals, but seems more problematic over the complex. $\endgroup$ – Bill Bradley Nov 5 '14 at 21:42

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