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This is a Quantitative Study study.

October 5, 2025Open Access

Heavy-tailed critical Galton--Watson processes with immigration

Key Points

The stationary distribution tail is regularly varying, revealing significant insights into the Galton-Watson process.
The study establishes a central limit theorem for partial sums, contributing a new understanding of the process's behavior.
The extremal index of the critical Galton-Watson process is found to be 0, indicating a unique stability characteristic.
The findings extend understanding of both finite mean immigration and varying stable laws in the tail behavior.

Abstract

Consider a critical Galton--Watson branching process with immigration, where the offspring distribution belongs to the domain of attraction of a (1 + α) -stable law with α (0, 1), and the immigration distribution either (i) has finite mean, or (ii) belongs to the domain of attraction of a β-stable law with β (α, 1). We show that the tail of the stationary distribution is regularly varying. We analyze the stationary process, determine its tail process, and establish a stable central limit theorem for the partial sums. The norming sequence is different from the one corresponding to the tail of the stationary law. In particular, the extremal index of the process is 0.

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Cite This Study

Kevei et al. (Thu,) studied this question.

synapsesocial.com/papers/68e25385d6d66a53c2474f43 https://doi.org/https://doi.org/10.48550/arxiv.2510.02004

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