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Population size of critical Galton-Watson processes under small deviations and infinite variance

Vladimir Vatutin, Elena Dyakonova

TL;DR

This work analyzes a critical Galton-Watson process with infinite offspring-variance, characterized by $f(s)=s+(1-s)^{1+\alpha}L(1-s)$ for $\alpha\in(0,1]$, and studies the distribution of the population size $Z(m)$ under the rare event that the distant-time population is small, encoded by the conditioning $\mathcal{H}(n,\varphi(n))$. By leveraging Yaglom-type limits, a reduced process $Z(m,n)$, and auxiliary asymptotics for $\mathcal{H}(n,\varphi(n))$, the authors derive explicit asymptotic Laplace transforms of $Z(m)$ across five regimes relating $m$ and $n$. The results reveal a rich set of conditional limit laws: a universal $(1+\lambda^{\alpha})^{-(1/\alpha+1)}$ form in the $m=o(n)$ regime; a theta-dependent family in the $m\sim \theta n$ regime; a near-terminal universal $(1+\lambda)^{-(\alpha+1)}$ form for $m=n-\psi(n)$; a convolutional series in the $n-m\sim y\varphi(n)$ case; and a beta-like $\alpha\int_0^1 x^{\alpha-1} e^{ -\lambda x } dx$ limit when the gap is $n-\chi(n)$. These findings extend Yaglom-type results to infinite-variance offspring laws and provide precise characterizations of how a small distant-time population emerges under various observational gaps, with implications for MRCA-distance scaling and conditional branching behavior.

Abstract

We study the evolution of the population size distribution of a critical Galton-Watson process with infinite variance of the offspring size of particles assuming that the population size is unusually small at the distant moment $n$ of observation.

Population size of critical Galton-Watson processes under small deviations and infinite variance

TL;DR

This work analyzes a critical Galton-Watson process with infinite offspring-variance, characterized by for , and studies the distribution of the population size under the rare event that the distant-time population is small, encoded by the conditioning . By leveraging Yaglom-type limits, a reduced process , and auxiliary asymptotics for , the authors derive explicit asymptotic Laplace transforms of across five regimes relating and . The results reveal a rich set of conditional limit laws: a universal form in the regime; a theta-dependent family in the regime; a near-terminal universal form for ; a convolutional series in the case; and a beta-like limit when the gap is . These findings extend Yaglom-type results to infinite-variance offspring laws and provide precise characterizations of how a small distant-time population emerges under various observational gaps, with implications for MRCA-distance scaling and conditional branching behavior.

Abstract

We study the evolution of the population size distribution of a critical Galton-Watson process with infinite variance of the offspring size of particles assuming that the population size is unusually small at the distant moment of observation.
Paper Structure (8 sections, 6 theorems, 127 equations)

This paper contains 8 sections, 6 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Population size
  4. The distribution of $Z(m)$ given $m=o(n)$ and $n\gg \varphi (n)$
  5. The distribution of $Z(m)$ if $m\sim \theta n$ for some $\theta \in (0,1)$
  6. The distribution of $Z(m)$ given $m=n-\psi (n)$ and $n\gg \psi (n)\gg \varphi (n)$
  7. The distribution of $Z(m)$ for $n-m\sim y\varphi (n)$, $y>0$
  8. The distribution of $Z(m)$ if $m=n-\chi(n)$ where $n\gg \varphi(n)\gg\chi (n)\geq 0$

Key Result

Theorem 1

Let conditions (MainAssump) be valid for $\alpha \in (0,1)$ and assumption (Def_phi) hold true. Then, for any $\lambda >0$ 1) if $m\rightarrow \infty$ and $m=o(n)$ then 2) if $m\sim \theta n$ for some $\theta \in (0,1)$ then 3) if $m=n-\psi (n),$ where $n\gg \psi (n)\gg \varphi (n),$ then 4) if $m=n-y\varphi (n),$ where $y\in (0,\infty ),$ then where $\Gamma (\cdot )$ is the standard Gamma-functio

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • Lemma 2