Asymptotic Behaviors for Critical Branching Processes with Immigration

TL;DR

This paper studies the long-term behavior of critical branching processes with immigration, providing estimates, large deviations, and bounds for the process and its maximum, enhancing understanding of their asymptotic properties.

Contribution

It introduces new asymptotic estimates and deviation bounds for critical branching processes with immigration, extending existing theoretical frameworks.

Findings

Derived estimation for the probability generating function of Z_n

Established large deviation principles for Z_{n+1}/Z_n

Obtained bounds for the maximum of Z_i over time

Abstract

In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\{Z_n, n\ge 0\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\max_{1\le i\le n} Z_i$ is obtained.