Lévy copulas: a probabilistic point of view
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There is a one-to-one correspondence between Lévy copulas and proper copulas. The correspondence relies on a relationship between Lévy copulas sitting on [0,+∞]^d and max-id distributions. The max-id distributions are defined with respect to a partial order that is compatible with the inclusion of sets bounded away from the origin. An important consequence of the result is the possibility to define parametric Lévy copulas as mirror images of proper parametric copulas. For example, proper Archimedean copulas are generated by functions that are Williamson d-transforms of the cdf of the radial component of random vectors with exchangeable distributions F_R. In contrast, the generators of Archimedean Lévy copulas are Williamson d-transforms of -log(1-F_R).
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