# A version of Herbert A. Simon's model with slowly fading memory and its connections to branching processes

## TL;DR

This paper introduces a recursive word-generation model inspired by Herbert A. Simon's work, analyzing its long-term behavior and connection to branching processes, revealing power-law and exponential decay regimes based on parameters.

## Contribution

It extends Simon's model by incorporating slowly fading memory and establishes links to branching processes, providing new insights into word frequency distributions.

## Key findings

- Proportion of words with exact repetitions converges as string length grows.
- Power-law decay in word frequency distribution when certain parameter conditions are met.
- Exponential decay occurs in the distribution under different parameter regimes.

## Abstract

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 $\in$ (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 $\alpha$ 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 $\ge$ 1, whose tail decays as a power function when 1 -- p > $\alpha$/(1 + $\alpha$), and exponentially fast when 1 -- p < $\alpha$/(1 + $\alpha$).