On selfsimilar Lévy Random Probabilities

We explore a class of random probabilities induced by the normalization of selfsimilar Lévy Random Measures–random measures whose probability laws are governed by stable one-sided Lévy distributions. Various statistical properties of these random probabilities are analyzed: (i) moment structure; (ii) auto-covariance structure; (iii) one-dimensional and multidimensional tail-probabilities; and (iv) behavior in limiting cases. For the Lévy–Smirnoff case—corresponding to the selfsimilarity index of order —an explicit analytic formula for the multidimensional probability density functions is derived. Last, a comparison between the class of selfsimilar Lévy Random Probabilities and the Dirichlet Random Probability (induced by the normalization of the Gamma Random Measure) is conducted, showing the former to be far more robust than the latter.