Limit theorems for the critical Galton-Watson processes with immigration stopped at zero

TL;DR

This paper investigates the behavior of a critical Galton-Watson process with immigration stopped at zero, providing precise estimations, tail probabilities, and establishing conditional limit theorems for the process.

Contribution

It introduces new estimations and limit theorems for the Galton-Watson process with immigration stopped at zero, advancing understanding of its long-term behavior.

Findings

Derived precise estimates of the generation function.

Analyzed the tail probability of the process's lifetime.

Established two conditional limit theorems for the process.

Abstract

In this paper, we consider a critical Galton-Watson branching process with immigration stopped at zero $\mathbf{W}$. Some precise estimation on the generation function of the $n$-th population are obtained, and the tail probability of the life period of $\mathbf{W}$ is studied. Based on above results, two conditional limit theorems for $\mathbf{W}$ are established.