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On new mechanisms leading to heavy-tailed distributions related to the ones of Yule-Simon. (English) Zbl 1457.60023
Summary: Scientists reinvent stochastic mechanisms leading to the emergence of a distribution discovered by H. A. Simon [Biometrika 42, 425–440 (1955; Zbl 0066.11201)], in the context of the study of word frequencies occurring in a textbook. Simon distributions are heavy-tailed as a result of a reinforcement mechanism that produced them, related to the modern notion of preferential attachment. The Simon distribution is a particular case of a distribution recently introduced, itself extending the Sibuya distribution. We exhibit some of the remarkable statistical properties of such a family of distributions, in particular the one of being discrete self-decomposable. Using this and after placing this problem in context, additional stochastic processes where such distributions naturally arise are investigated, in particular a Markov chain model with catastrophes.

MSC:
60E05 Probability distributions: general theory
60E07 Infinitely divisible distributions; stable distributions
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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