Chance Mechanisms Involving Sibuya Distribution and its Relatives
Résumé
The two-parameters generalized Sibuya discrete distributions capture the essence of random phenomena presenting large probability mass near the lower bound of its support balanced with heavy-tails in their deep upper bound. They are heavy-tailed as a result of the reinforcement mechanism that produced them, related to the modern notion of preferential attachment. We describe stochastic mechanisms (chiefly Markov chains) leading to the emergence of such distributions, starting with the particular case of the one-parameter Simon distribution appearing in the context of word frequencies occurring in a textbook. We exhibit some of the remarkable statistical properties of the generalized Sibuya distributions. A second related two-parameters Sibuya family is investigated in the same spirit: the class of scaled Sibuya distributions.