Functional limit theorems for Galton-Watson processes with inhomogeneous immigration towards a deterministic limit function
Abstract
We study the asymptotic behavior of a sequence of Galton-Watson processes with inhomogeneous immigration when the limit of the means of the offspring distributions is equal to 1. Under a growth condition on the variances of the offspring distributions, assuming the locally uniform convergence of the expectation functions of properly scaled immigration processes towards a càdlàg function, and the locally uniform convergence of the mean absolute deviation functions of the scaled immigration processes in question towards zero, we establish a functional limit theorem for the sequence of Galton-Watson processes with inhomogeneous immigration in question towards a deterministic limit process. We also provide some applications, among others, we show that a recent result of Rahimov and Sharipov (2024) on a functional limit theorem for a sequence of nearly critical Galton-Watson processes with dependent immigration can be recovered using our result.
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BibTeX
@misc{barczy2026functional,
title = {{Functional limit theorems for Galton-Watson processes with inhomogeneous immigration towards a deterministic limit function}},
author = {Matyas Barczy and Dániel Bezdány},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39771},
eprint = {2610.39771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39771},
}
RIS
TY - UNPB
TI - Functional limit theorems for Galton-Watson processes with inhomogeneous immigration towards a deterministic limit function
AU - Matyas Barczy
AU - Dániel Bezdány
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39771
UR - https://arxiv.org/abs/2610.39771
AB - We study the asymptotic behavior of a sequence of Galton-Watson processes with inhomogeneous immigration when the limit of the means of the offspring distributions is equal to 1. Under a growth condition on the variances of the offspring distributions, assuming the locally uniform convergence of the expectation functions of properly scaled immigration processes towards a càdlàg function, and the locally uniform convergence of the mean absolute deviation functions of the scaled immigration processes in question towards zero, we establish a functional limit theorem for the sequence of Galton-Watson processes with inhomogeneous immigration in question towards a deterministic limit process. We also provide some applications, among others, we show that a recent result of Rahimov and Sharipov (2024) on a functional limit theorem for a sequence of nearly critical Galton-Watson processes with dependent immigration can be recovered using our result.
ER -