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lrnv committed Feb 14, 2024
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Expand Up @@ -88,7 +88,6 @@ In this package, we implemented it through the [`WilliamsonGenerator`](@ref) cla

`WilliamsonGenerator(X::UnivariateRandomVariable, d)`.


This function computes the Williamson d-transform of the provided random variable $X$ using the [`WilliamsonTransforms.jl`](https://github.com/lrnv/WilliamsonTransforms.jl) package. See [williamson1955multiply, mcneil2009](@cite) for the literature.

!!! warn "`max_monotony` of Williamson generators"
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## [Inverse Williamson d-transform](@id w_trans_section)


The Williamson d-transform is a bijective transformation[^1] from the set of positive random variables to the set of generators. It therefore has an inverse transformation (called, surprisingly, the inverse Williamson $d$-transform) that construct the positive random variable *R* from a generator $\phi$.

[^1]:
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As obvious from the definition of the Williamson transform, using a discrete distribution produces piecewise-linear generators, where the number of pieces is dependent on the order of the transformation.


## Archimedean copulas

Let's first define formally archimedean copulas:
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