Abstract
We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in Bertoin (J Stat Phys 176(3):679–691, 2019) in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in Bertoin (J Stat Phys 176(3):679–691, 2019). Finally, by superposing mutations to the branching process, we propose a model which leads to the two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon (Biometrika 42(3/4):425–440, 1955) on limiting word frequencies.
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Notes
- 1.
Although the probability \(\mathbb {P}(X_{\theta , \rho }=1)\) can easily be computed in terms of an incomplete Gamma function, the calculations needed to determine \(\mathbb {P}(X_{\theta , \rho }=k)\) for k ≥ 2 become soon intractable.
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Baur, E., Bertoin, J. (2021). On a Two-Parameter Yule-Simon Distribution. In: Chaumont, L., Kyprianou, A.E. (eds) A Lifetime of Excursions Through Random Walks and Lévy Processes. Progress in Probability, vol 78. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83309-1_4
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