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


Bertoin, Jean (2019). A version of Herbert A. Simon’s model with slowly fading memory and its connections to branching processes. Journal of Statistical Physics, 176(3):679-691.

Abstract

Construct recursively a long string of words $w_1$,…$w_n$, 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 $j^α$ for $j=1$,…,$k$. We show that the proportion of distinct words occurring exactly $ℓ$ times converges as the length $n$ of the string goes to infinity to some probability mass function in the variable $ℓ≥1$, whose tail decays as a power function when $p<1/(1+α)$, and exponentially fast when $p>1/(1+α)$.

Abstract

Construct recursively a long string of words $w_1$,…$w_n$, 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 $j^α$ for $j=1$,…,$k$. We show that the proportion of distinct words occurring exactly $ℓ$ times converges as the length $n$ of the string goes to infinity to some probability mass function in the variable $ℓ≥1$, whose tail decays as a power function when $p<1/(1+α)$, and exponentially fast when $p>1/(1+α)$.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistical and Nonlinear Physics
Physical Sciences > Mathematical Physics
Uncontrolled Keywords:Mathematical Physics, Statistical and Nonlinear Physics
Language:English
Date:1 August 2019
Deposited On:28 Jun 2019 09:20
Last Modified:29 Jul 2020 10:54
Publisher:Springer
ISSN:0022-4715
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s10955-019-02316-1

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