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Local Limit Theorem for Branching Processes in Varying Environments
Abstract
Let Zi,n be the population at time n descended from one particle at time i in a branching process in a varying environment. Under two-sided bounds on products of offspring means, an integrable uniform second-moment tail condition, and nondegeneracy and unit positive-support gcd of all weak limits, we prove ℙ(Zi,n=k)≤ C(n−i+1)−2 uniformly in 0≤ i<n and k≥1, and obtain a local limit theorem as n−i,k→∞ with k≤ L0(n−i), without third-moment assumptions. These results extend the classical local probability estimates for critical Galton--Watson processes to varying environments.
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BibTeX
@misc{xia2026local,
title = {{Local Limit Theorem for Branching Processes in Varying Environments}},
author = {Zhiyang Xia and Mei Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01455},
eprint = {2611.01455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01455},
}
RIS
TY - UNPB
TI - Local Limit Theorem for Branching Processes in Varying Environments
AU - Zhiyang Xia
AU - Mei Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01455
UR - https://arxiv.org/abs/2611.01455
AB - Let Z_i,n be the population at time n descended from one particle at time i in a branching process in a varying environment. Under two-sided bounds on products of offspring means, an integrable uniform second-moment tail condition, and nondegeneracy and unit positive-support gcd of all weak limits, we prove ℙ(Z_i,n=k)≤ C(n−i+1)^−2 uniformly in 0≤ i<n and k≥1, and obtain a local limit theorem as n−i,k→∞ with k≤ L_0(n−i), without third-moment assumptions. These results extend the classical local probability estimates for critical Galton--Watson processes to varying environments.
ER -