Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reinforced Galton--Watson processes III: Empirical offspring distributions

Jean Bertoin, Bastien Mallein

TL;DR

This work extends Galton–Watson trees by incorporating reinforcement through a memory parameter $q\in(0,1)$, causing offspring counts to be echoed along ancestral lineages via the empirical distribution $\boldsymbol{\mu}_v$. The authors develop a large deviations framework for reinforced sampling, establishing exponential concentration of empirical offspring distributions around a reinforced size-biased law $\overline{\boldsymbol{\nu}}_q$ and deriving precise survival or evanescence criteria through rate-function analyses. A central contribution is the reinforced Sanov theorem and its Fenchel–Legendre representation, enabling explicit characterizations of typical and atypical genealogies under reinforcement. The results connect to classical GW theory via spinal decompositions and extend them to multi-type and reinforced settings, providing practical criteria and explicit constructions (e.g., via Lambert $W$) for survival and persistence. Overall, the paper advances understanding of how memory effects alter the long-run composition and viability of populations modeled by branching processes, with potential implications for population genetics and network growth models.

Abstract

Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution depends on the empirical offspring distribution of each individual along its ancestral lineage. We are interested here in asymptotic properties of the empirical distributions observed in the population, such as concentration, evanescence and persistence. For this, we incorporate tools from the theory of large deviations to our preceding analysis [arXiv:2306.02476,arXiv:2310.19030].

Reinforced Galton--Watson processes III: Empirical offspring distributions

TL;DR

This work extends Galton–Watson trees by incorporating reinforcement through a memory parameter , causing offspring counts to be echoed along ancestral lineages via the empirical distribution . The authors develop a large deviations framework for reinforced sampling, establishing exponential concentration of empirical offspring distributions around a reinforced size-biased law and deriving precise survival or evanescence criteria through rate-function analyses. A central contribution is the reinforced Sanov theorem and its Fenchel–Legendre representation, enabling explicit characterizations of typical and atypical genealogies under reinforcement. The results connect to classical GW theory via spinal decompositions and extend them to multi-type and reinforced settings, providing practical criteria and explicit constructions (e.g., via Lambert ) for survival and persistence. Overall, the paper advances understanding of how memory effects alter the long-run composition and viability of populations modeled by branching processes, with potential implications for population genetics and network growth models.

Abstract

Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution depends on the empirical offspring distribution of each individual along its ancestral lineage. We are interested here in asymptotic properties of the empirical distributions observed in the population, such as concentration, evanescence and persistence. For this, we incorporate tools from the theory of large deviations to our preceding analysis [arXiv:2306.02476,arXiv:2310.19030].

Paper Structure

This paper contains 12 sections, 16 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some definition and a many-to-one formula
  4. Basic background from large deviations theory
  5. The benchmark case of Galton--Watson trees
  6. Weak persistence in a two-type Galton-Watson tree
  7. Reinforced sampling
  8. Reinforced Sanov's theorem
  9. Fenchel-Legendre identification of the rate function
  10. Reinforced Galton--Watson processes
  11. A sufficient condition for survival
  12. Funding.

Key Result

Lemma 2.4

Take any $(x_0,\ldots,x_{n-1}) \in (S\setminus\{0\})^n$ for some $n\geq 1$ and set $\boldsymbol{\mu} = \frac{1}{n} \sum_{j=0}^{n-1} \boldsymbol{\delta}_{x_j}$. Using the notation E:rhoell, we have In particular, for any probability measure $\boldsymbol{\varrho}\in \mathcal{P}_S$ such that the measure $n\boldsymbol{\varrho}$ is integer-valued, there is the identity

Figures (2)

  • Figure 1: A representation of the tree $T$ constructed in Example \ref{['expl']} encoded with its Ulam--Harris notation. The vertices of the original binary tree are represented as circles.
  • Figure 2: Graphs of $H(\boldsymbol{\varrho} | q\boldsymbol{\varrho} + (1-q) \boldsymbol{\nu})$ (blue) and of $\Lambda_q^*(\boldsymbol{\varrho})$ (red) for $\boldsymbol{\varrho}=(p,1-p)$ with $p\in[0,1/2]$ in Example \ref{['E:es']}. The function $H(\cdot | q\cdot + (1-q)\boldsymbol{\nu})$ appears as a very good approximation to $\Lambda_q^*$.

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4: Many-to-one formula
  • Remark 2.5
  • proof
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 34 more