Reinforced Galton–Watson processes I: Malthusian exponents

In a reinforced Galton–Watson process with reproduction law ν$$ \boldsymbol{\nu} $$ and memory parameter q∈(0,1)$$ q\in \left(0,1\right) $$, the number of children of a typical individual either, with probability q$$ q $$, repeats that of one of its forebears picked uniformly at random, or, with com...

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Published in:Random structures & algorithms Vol. 65; no. 2; pp. 387 - 410
Main Authors: Bertoin, Jean, Mallein, Bastien
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.09.2024
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Subjects:
Branching (mathematics)
Combinatorics
Galton–Watson process
Malthusian growth exponent
Mathematics
Probability
singularity analysis of generating functions
Stochastic processes
stochastic reinforcement
transport equation
Transport equations
Galton-Watson process
stochastic reinforcement
transport equation
Malthusian growth exponent
singularity analysis of generating functions
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:In a reinforced Galton–Watson process with reproduction law ν$$ \boldsymbol{\nu} $$ and memory parameter q∈(0,1)$$ q\in \left(0,1\right) $$, the number of children of a typical individual either, with probability q$$ q $$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability 1−q$$ 1-q $$, is given by an independent sample from ν$$ \boldsymbol{\nu} $$. We estimate the average size of the population at a large generation, and in particular, we determine explicitly the Malthusian growth rate in terms of ν$$ \boldsymbol{\nu} $$ and q$$ q $$. Our approach via the analysis of transport equations owes much to works by Flajolet and co‐authors.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21219