Left tail asymptotics for the density of supercritical branching processes in random environment
Abstract
Let (Zn)n≥ 0 be a supercritical branching process in a random environment in which every individual produces at least one offspring, and let W be the almost sure limit of the normalized Zn. Write p1 for the (random) probability that an individual has exactly one child and m0 for the mean offspring number in a given environment. Assume that p1 is positive almost surely and bounded away from $1$, that m0 is bounded away from $1$ and from infinity together with a mild moment condition, and that log m0 is non-lattice. Let a0>0 be the unique solution of the equation 𝔼[p1 m0a0]=1. Under these assumptions, we determine the precise left tail asymptotics of the annealed law of W: the distribution function F(x)=ℙ(W≤ x) satisfies F(x)∼ D xa0 as x↓ 0 with a constant D∈(0,∞) admitting an explicit representation, and the annealed density w of W, which we show exists and can be chosen continuous on (0,∞), satisfies w(x)∼ G xa0−1 as x↓ 0 with G=a0 D>0. These results extend the classical power-law left tail behavior of the martingale limit of a Galton--Watson process in the literature to the random environment setting, where the exponent a0 is determined by the averaged environmental law rather than by a fixed offspring distribution.
Related
- Limit theorems for supercritical branching processes in Markovian environment with linear fractional offspring distributions
- Weighted moments and sharp convergence rates for supercritical multitype branching processes
- Supercritical Branching Random Walks on Nested Lattices
- Continuous-state branching processes with Lévy-Khintchine drift-interaction: Laplace duality and Fellerian extensions
- Quenched First-Passage Asymptotics for Branching Random Walk on a Hamming Cube
BibTeX
@misc{chen2026left,
title = {{Left tail asymptotics for the density of supercritical branching processes in random environment}},
author = {Panyi Chen and Mei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25848},
eprint = {2610.25848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25848},
}
RIS
TY - UNPB
TI - Left tail asymptotics for the density of supercritical branching processes in random environment
AU - Panyi Chen
AU - Mei Zhang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25848
UR - https://arxiv.org/abs/2610.25848
AB - Let (Z_n)_n≥ 0 be a supercritical branching process in a random environment in which every individual produces at least one offspring, and let W be the almost sure limit of the normalized Z_n. Write p_1 for the (random) probability that an individual has exactly one child and m_0 for the mean offspring number in a given environment. Assume that p_1 is positive almost surely and bounded away from $1$, that m_0 is bounded away from $1$ and from infinity together with a mild moment condition, and that log m_0 is non-lattice. Let a_0>0 be the unique solution of the equation 𝔼[p_1 m_0^a_0]=1. Under these assumptions, we determine the precise left tail asymptotics of the annealed law of W: the distribution function F(x)=ℙ(W≤ x) satisfies F(x)∼ D x^a_0 as x↓ 0 with a constant D∈(0,∞) admitting an explicit representation, and the annealed density w of W, which we show exists and can be chosen continuous on (0,∞), satisfies w(x)∼ G x^a_0−1 as x↓ 0 with G=a_0 D>0. These results extend the classical power-law left tail behavior of the martingale limit of a Galton--Watson process in the literature to the random environment setting, where the exponent a_0 is determined by the averaged environmental law rather than by a fixed offspring distribution.
ER -