A version of Herbert A. Simon's model with slowly fading memory and its connections to branching processes
Résumé
Construct recursively a long string of words w1. .. wn, such that at each step k, w k+1 is a new word with a fixed probability p ∈ (0, 1), and repeats some preceding word with complementary probability 1 − p. More precisely, given a repetition occurs, w k+1 repeats the j-th word with probability proportional to j α for j = 1,. .. , k. We show that the proportion of distinct words occurring exactly times converges as the length n of the string goes to infinity to some probability mass function in the variable ≥ 1, whose tail decays as a power function when 1 − p > α/(1 + α), and exponentially fast when 1 − p < α/(1 + α).
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