A Version of Herbert A. Simon’s Model with Slowly Fading Memory and Its Connections to Branching Processes

  • Jean BertoinEmail author


Construct recursively a long string of words \(w_1,\ldots w_n,\) such that at each step k\(w_{k+1}\) is a new word with a fixed probability \(p\in (0,1),\) and repeats some preceding word with complementary probability \(1-p.\) More precisely, given a repetition occurs, \(w_{k+1}\) repeats the jth word with probability proportional to \(j^{\alpha }\) for \(j=1,\ldots , k.\) We show that the proportion of distinct words occurring exactly \(\ell \) times converges as the length n of the string goes to infinity to some probability mass function in the variable \(\ell \ge 1,\) whose tail decays as a power function when \(p<1/(1+\alpha ),\) and exponentially fast when \(p>1/(1+\alpha ).\)


Yule–Simon model Preferential attachment Memory Continuous state branching process Crump–Mode–Jagers branching process Heavy tail distributions 

Mathematics Subject Classification

60J85 05C85 



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

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