On a Two-Parameter Yule-Simon Distribution

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.