On the Kolmogorov Distance of Max-Stable Distributions

In this contribution, we derive explicit bounds on the Kolmogorov distance for multivariate max-stable distributions with Fr\'echet margins. We formulate those bounds in terms of (i) Wasserstein distances between de Haan representers, (ii) total variation distances between spectral/angular measures - removing the dimension factor from earlier results in the canonical sphere case - and (iii) discrepancies of the Psi-functions in the inf-argmax decomposition. Extensions to different margins and Archimax/clustered Archimax copulas are further discussed. Examples include logistic, comonotonic, independent and Brown-Resnick models.