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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| Veröffentlicht in: | Random structures & algorithms Jg. 65; H. 2; S. 387 - 410 |
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John Wiley & Sons, Inc
01.09.2024
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| Abstract | 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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| AbstractList | 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. In a reinforced Galton–Watson process with reproduction law and memory parameter , the number of children of a typical individual either, with probability , repeats that of one of its forebears picked uniformly at random, or, with complementary probability , is given by an independent sample from . 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 and . Our approach via the analysis of transport equations owes much to works by Flajolet and co‐authors. In a reinforced Galton-Watson process with reproduction law $\boldsymbol{\nu}$ and memory parameter $q\in(0,1)$, the number of children of a typical individual either, with probability $q$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability $1-q$, is given by an independent sample from $\boldsymbol{\nu}$. 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 $\boldsymbol{\nu}$ and $q$. Our approach via the analysis of transport equations owns much to works by Flajolet and co-authors. |
| Author | Mallein, Bastien Bertoin, Jean |
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| Keywords | Galton-Watson process stochastic reinforcement transport equation Malthusian growth exponent singularity analysis of generating functions |
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| Snippet | 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... In a reinforced Galton–Watson process with reproduction law and memory parameter , the number of children of a typical individual either, with probability ,... In a reinforced Galton-Watson process with reproduction law $\boldsymbol{\nu}$ and memory parameter $q\in(0,1)$, the number of children of a typical individual... |
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| SubjectTerms | Branching (mathematics) Combinatorics Galton–Watson process Malthusian growth exponent Mathematics Probability singularity analysis of generating functions Stochastic processes stochastic reinforcement transport equation Transport equations |
| Title | Reinforced Galton–Watson processes I: Malthusian exponents |
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