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lrnv committed Sep 22, 2024
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2 changes: 1 addition & 1 deletion docs/src/archimedean/generalities.md
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Expand Up @@ -93,7 +93,7 @@ In this package, we implemented it through the [`WilliamsonGenerator`](@ref) cla

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"
!!! warning "`max_monotony` of Williamson generators"
The $d$-transform of a positive random variable is $d$-monotonous but not $k$-monotonous for any $k > d$. Its max monotony is therefore $d$. This has a few implications, one of the biggest one is that the $d$-variate Archimedean copula that corresponds has no density.

More genrally, if you want your Archimedean copula to have a density, you have to use a generator that is more-monotonous that the dimension of your model.
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2 changes: 1 addition & 1 deletion src/SklarDist.jl
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Expand Up @@ -11,7 +11,7 @@ Constructor
This function allows to construct a random vector specified, through the Sklar Theorem, by its marginals and its copula separately. See [Sklar's theorem](https://en.wikipedia.org/wiki/Copula_(probability_theory)#Sklar's_theorem):
!!!theorem "Theorem (Sklar 1959):"
!!! theorem "Theorem (Sklar 1959):"
For every random vector ``\\bm X``, there exists a copula ``C`` such that
``\\forall \\bm x\\in \\mathbb R^d, F(\\bm x) = C(F_{1}(x_{1}),...,F_{d}(x_{d})).``
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