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Functional limit theorems for Galton--Watson processes with inhomogeneous immigration

  • Matyas Barczy
  • Dániel Bezdány
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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 less than $1$ or equal to $1$. Under growth conditions on the expected values of the immigration distributions and the variances of the offspring distributions, and assuming the weak convergence of properly scaled immigration processes towards a non-negative stochastic process 𝒴 with càdlàg or continuous sample paths, we establish functional limit theorems for the sequence of Galton--Watson processes with inhomogeneous immigration in question. The limit stochastic processes can be represented as a constant multiple or an integral functional of 𝒴 .

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Cite

BibTeXRIS
BibTeX
@misc{barczy2026functional,
  title = {{Functional limit theorems for Galton--Watson processes with inhomogeneous immigration}},
  author = {Matyas Barczy and Dániel Bezdány},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.18503},
  eprint = {2609.18503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18503},
}
RIS
TY  - UNPB
TI  - Functional limit theorems for Galton--Watson processes with inhomogeneous immigration
AU  - Matyas Barczy
AU  - Dániel Bezdány
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.18503
UR  - https://arxiv.org/abs/2609.18503
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 less than $1$ or equal to $1$. Under growth conditions on the expected values of the immigration distributions and the variances of the offspring distributions, and assuming the weak convergence of properly scaled immigration processes towards a non-negative stochastic process 𝒴 with càdlàg or continuous sample paths, we establish functional limit theorems for the sequence of Galton--Watson processes with inhomogeneous immigration in question. The limit stochastic processes can be represented as a constant multiple or an integral functional of 𝒴 .
ER  -