Heavy-tailed critical Galton--Watson processes with immigration

TL;DR

This work analyzes a critical Galton–Watson process with immigration where offspring tails lie in the domain of attraction of a $(1+\alpha)$-stable law and immigration is either finite-mean or in the domain of a $\beta$-stable law with $\beta\in(\alpha,1)$. It proves that the stationary distribution is regularly varying with explicit constants in both immigration regimes, and it derives the tail process showing a forward Pareto structure that leads to a zero extremal index. A stable central limit theorem with nonstandard normalization is established for partial sums, highlighting pathological extremal dependence in this critical, heavy-tailed setting. The results extend heavy-ta-tail time-series theory to branching processes with immigration and illuminate how immigration regime governs tail behavior, extremes, and sums. Techniques combine Tauberian analysis of generating functions, tail-process methods, and random-sum limit results.

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 $α\in (0,1)$, and the immigration distribution either (i) has finite mean, or (ii) belongs to the domain of attraction of a $β$-stable law with $β\in (α, 1)$. 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 $0$.

Theorems & Definitions (11)

• Corollary 1
• Lemma 2
• Theorem 3
• Example : Power-fractional offspring distribution or theta-branching
• Theorem 4
• Lemma 5
• Theorem 6
• Corollary 7
• proof : Proof of Lemma \ref{['lemma:T-tail']}
• Lemma 8